Multidimensional Knapsack Problem Research Articles

QoS-Aware Web Service Selection with Internal Complementarity

Service composition is a key enabling technology in service-oriented computing for developing versatile applications by integrating various existing interoperable services. Although actively studied, most existing works on service composition neglect the existence of complementarity among candidate services within a service class, so-called internal complementarity. In fact, complementary candidate services within a service class can be composed to form a composite candidate service which may yield better service utility than that provided by any existing candidate service within that service class. This work focuses on web service composition where internal complementarity is more likely to happen. Specifically, we aim at addressing the problem of QoS-aware web service selection with internal complementarity (WSS-IC). We first transform this problem into a multi-choice multi-dimensional knapsack problem (MMKP) and prove such a transformation per se has non-polynomial time complexity in the worse case.Then, we perform complexity analysis to demonstrate that existing approaches to MMKPs are not computationally feasible to resolve QoS-aware WSS-IC. This fact motivates us to propose an iteratively improving framework for deriving the solution iteration by iteration while taking into account both solution structure and QoS constraints. At each iteration, the current solution gets improved by solving a disjunctively constrained knapsack problem. To verify the effectiveness of the proposed framework, two heuristic approaches are implemented under this framework. Experimental results demonstrate that our approaches outperform the compared methods in terms of both solution quality and computation time.

Human–Robot Collaborative Site Inspection Under Resource Constraints

This paper is on human-robot collaborative site inspection and target classification. We consider the realistic case that human visual performance is imperfect (depending on the sensory input quality), and that the robot has constraints in communication with human (e.g., limited chances for query, poor channel quality). The robot has limited onboard motion and communication energy and operates in realistic channel environments experiencing path loss, shadowing, and multipath. We then show how to co-optimize motion, sensing, and human queries. Given a probabilistic assessment of human visual performance and a probabilistic channel prediction, we pose the co-optimization as multiple-choice multidimensional knapsack problems. We then propose a linear program-based efficient near-optimal solution, mathematically characterize the optimality gap, showing it to be very small, and mathematically characterize properties of the optimum solution. We then comprehensively validated the proposed approach with extensive real human data (from Amazon MTurk) and real channel data (from downtown San Francisco), confirming that the proposed approach significantly outperforms benchmark methodologies.

Bridging k-sum and CVaR optimization in MILP

Mixed-Integer Linear Programming models often optimize the sum, or average, of different outcomes. In a deterministic setting, each outcome may be associated with an agent, for example a customer, an employee, or a time period. In a stochastic setting, each outcome may be associated with a discrete scenario. The average approach optimizes the overall efficiency of the solution, but neglects the possible unfair distribution of outcomes among agents or the risk of very bad scenarios. In this paper, we exploit the analogies of the two settings to derive a common optimization paradigm bridging the gap between k-sum optimization in the deterministic setting and Conditional Value-at-Risk optimization in the stochastic setting. We show that the proposed paradigm satisfies properties that make it an attractive criterion. To illustrate the proposed paradigm, we apply it to the multidimensional knapsack problem and the p-median/p-center problem.

A pruned Pareto set for multi-objective optimisation problems via particle swarm and simulated annealing

A Pareto optimal set, which is obtained from solving multi-objective optimisation problems, usually contain a large number of optimal solutions. This situation poses a challenge for decision makers in choosing a suitable solution from a large number of overlapping and complex Pareto solutions. This paper proposes a new procedure for solving multi-objective optimisation problems by reducing the size of the Pareto set. The procedure is divided into two major stages. In the first stage, the multi-objective simulated annealing algorithm is used to solve a multi-objective optimisation problem by constructing the Pareto optimal set. In the second stage, the automatic clustering algorithm is used to prune the Pareto set. This procedure is implemented to solve two multi-objective optimisation problems, namely, the 0/1 multi-objective multi-dimensional knapsack problem and the multi-objective inventory system. The procedure enables the decision maker to select an appropriate solution efficiently.

BMNABC: Binary Multi-Neighborhood Artificial Bee Colony for High-Dimensional Discrete Optimization Problems

Many meta-heuristic algorithms have been proposed to solve continuous optimization problems. Hence, researchers have applied various techniques to change these algorithms for discrete search spaces. Artificial bee colony (ABC) algorithm is one of the well-known algorithms for real search spaces. ABC has a good ability in exploration but it is weak in exploitation. Several binary versions of ABC have been proposed so far. Since the methods are based on the standard ABC, they have the disadvantage of ABC. In this article, a new binary ABC called binary multi-neighborhood ABC (BMNABC) has been introduced to enhance the exploration and exploitation abilities in the phases of ABC. BMNABC applies the near and far neighborhood information with a new probability function in the first and second phases. A more conscious search than the standard ABC is done in the third phase for those solutions which have been not improved in the previous phases. The performance of algorithm has been evaluated by low- and high-dimensional functions and the 0-1 multidimensional knapsack problems. The proposed method has been compared with state-of-the-art algorithms. The results showed that BMNABC had a better performance in terms of solution accuracy and convergence speed.

Minimal Cost Server Configuration for Meeting Time-Varying Resource Demands in Cloud Centers

We consider the minimal cost server configuration for meeting resource demands over multiple time slots. Specifically, there are some heterogeneous servers. Each server is specified by a cost, certain amounts of several resources, and an active interval, i.e., the time interval that the server is planed to work. There are different overall demands for each type of resource over different time slots. A feasible solution is a set of servers such that at any time slot, the resources provided by the selected servers are at least their corresponding demands. Notice that, a selected server can not provide resources for the time slots out of its active interval. The total cost of the solution is the summation of the costs of all selected servers. The goal is to find a feasible solution with minimal total cost. This problem is proved to be NP-hard due to a reduction from the multidimensional knapsack problem (MKP), which is a well-known NP-hard combinational optimization problem. To solve our problem, we present a randomized approximation algorithm called partial rounding algorithm ( $\mathcal {PRA}$ ), which guarantees $O\left(\log \left(KT \right) \right)$ -approximation, i.e., $\eta \;\log \left(KT \right)$ -approximation, where $K$ is the number of kinds of resources, $T$ is the number of time slots, and $\eta$ is a positive constant. Furthermore, to minimize $\eta$ as much as possible, we propose a varied Chernoff bound and apply it in $\mathcal {PRA}$ . We perform extensive experiments with random inputs and a specific application input. The results show that $\mathcal {PRA}$ with our varied Chernoff conclusion can find solutions closing to the optimal one.

Two-stage solution-based tabu search for the multidemand multidimensional knapsack problem

The multidemand multidimensional knapsack problem (MDMKP) is a significant generalization of the popular multidimensional knapsack problem with relevant applications. In this work we investigate for the first time how solution-based tabu search can be used to solve this computationally challenging problem. For this purpose, we propose a two-stage search algorithm, where the first stage aims to locate a promising hyperplane within the whole search space and the second stage tries to find improved solutions by exploring the reduced subspace defined by the hyperplane. Computational experiments on 156 benchmark instances commonly used in the literature show that the proposed algorithm competes favorably with the state-of-the-art results. We analyze several key components of the algorithm to highlight their impacts on the performance of the algorithm.

Complexity indices for the multidimensional knapsack problem

In this article, the concept of conditioning in integer programming is extended to the concept of a complexity index. A complexity index is a measure through which the execution time of an exact algorithm can be predicted. We consider the multidimensional knapsack problem with instances taken from the OR-library and MIPLIB. The complexity indices we developed correspond to the eigenvalues of a Dikin matrix placed in the center of a polyhedron defined by the constraints of the problem relaxed from its binary variable formulation. The lower and higher eigenvalues, as well as their ratio, which we have defined as the slenderness, are used as complexity indices. The experiments performed show a good linear correlation between these indices and a low execution time of the Branch and Bound algorithm using the standard version of CPLEX® 12.2. The correlation coefficient obtained ranges between 39 and 60% for the various data regressions, which proves a medium to strong correlation.

The robust multiple-choice multidimensional knapsack problem

The multiple-choice multidimensional knapsack problem (MMKP) assumes n sets composed of mutually exclusive items. The goal is to select exactly one item per set, maximizing the overall utility, without violating a family of knapsack constraints. Motivated by recent applications of the MMKP to complex system reliability and quality of service management problems, we propose a robust version. More specifically, we relinquish the assumption that the problem parameters are deterministically known by limiting their values to a pre-specified uncertainty set. Depending on the structure of the variance−covariance matrix used to model the uncertainty, we identify four different cases, leading to robust formulations characterized by second order cone programs. We show how each of these programs is transformed into an equivalent linear program, implying that the use of a robust formulation for the MMKP comes with no extra computational complexity. Finally, using a novel matheuristic designed for the MMKP, we shed lights on the trade-off between the “price of robustness,” i.e., how much worse the objective function value of a robust solution is, compared with the deterministic one, and the “reliability,” i.e., the probability that a robust solution will lead to a feasible scenario for an arbitrary realization of the uncertain parameters.

THE CONCEPT OF GUARANTEED SOLUTION THROUGH THE FUNCTIONAL FOR MUTIDIMENTIONAL KNAPSACK PROBLEM AND METHODS OF ITS CONSTRUCTION

Contex. The problem of constructing a guaranteed suboptimal (approximate) solution with respect to a functional in one-dimensional and multidimensional knapsack problems is considered. The object of the study was a model with an increment of the coefficients of the objective function. Objective. The methods of constructing guaranteed suboptimal solution through the functional in one-dimensional and multidimensional knapsack problem has been developed. That is it is necessary to find such minimal changes coefficient of the objective function in the set of integer intervals so that the solution found guarantees the value of the functional not less than the predetermined value. Method. The concept of guaranteed solution and guaranteed suboptimal solution relative to the objective function in the satchel problem is introduced. It is necessary to find such minimal changes coefficient of the objective function in the set of integer intervals so that the solution found guarantees the value of the functional not less than the predetermined value. Such kind of solution we name as guaranteed solution through the functional for one-dimensional and multidimensional knapsack problem. The methods of their construction has been developed. A software package was developed to find these solutions and numerous computational experiments were performed on random large-dimensional problems. Results. The algorithm of constructing guaranteed suboptimal solution through the functional in one-dimensional and multidimensional knapsack problem has been developed. Conclusions. A software package was developed to find the concept of guaranteed solution and guaranteed suboptimal solutions and numerous computational experiments were performed on random large-dimensional problems.

Stabilized branch-and-price algorithms for vector packing problems

This paper considers packing and cutting problems in which a packing/cutting pattern is constrained independently in two or more dimensions. Examples are restrictions with respect to weight, length, and value. We present branch-and-price algorithms to solve these vector packing problems (VPPs) exactly. The underlying column-generation procedure uses an extended master program that is stabilized by (deep) dual-optimal inequalities. While some inequalities are added to the master program right from the beginning (static version), other violated dual-optimal inequalities are added dynamically. The column-generation subproblem is a multidimensional knapsack problem, either binary, bounded, or unbounded depending on the specific master problem formulation. Its fast resolution is decisive for the overall performance of the branch-and-price algorithm. In order to provide a generic but still efficient solution approach for the subproblem, we formulate it as a shortest path problem with resource constraints (SPPRC), yielding the following advantages: (i) Violated dual-optimal inequalities can be identified as a by-product of the SPPRC labeling approach and thus be added dynamically; (ii) branching decisions can be implemented into the subproblem without deteriorating its resolution process; and (iii) larger instances of higher-dimensional VPPs can be tackled with branch-and-price for the first time. Extensive computational results show that our branch-and-price algorithms are capable of solving VPP benchmark instances effectively.

Network Pollution Games

The problem of pollution control has been mainly studied in the environmental economics literature where the methodology of game theory is applied for the pollution control. To the best of our knowledge this is the first time this problem is studied from the computational point of view. We introduce a new network model for pollution control and present two applications of this model. On a high level, our model comprises a graph whose nodes represent the agents, which can be thought of as the sources of pollution in the network. The edges between agents represent the effect of spread of pollution. The government who is the regulator, is responsible for the maximization of the social welfare and sets bounds on the levels of emitted pollution in both local areas as well as globally in the whole network. We first prove that the above optimization problem is NP-hard even on some special cases of graphs such as trees. We then turn our attention on the classes of trees and planar graphs which model realistic scenarios of the emitted pollution in water and air, respectively. We derive approximation algorithms for these two kinds of networks and provide deterministic truthful and truthful in expectation mechanisms. In some settings of the problem that we study, we achieve the best possible approximation results under standard complexity theoretic assumptions. Our approximation algorithm on planar graphs is obtained by a novel decomposition technique to deal with constraints on vertices. We note that no known planar decomposition techniques can be used here and our technique can be of independent interest. For trees we design a two level dynamic programming approach to obtain an FPTAS. This approach is crucial to deal with the global pollution quota constraint. It uses a special multiple choice, multi-dimensional knapsack problem where coefficients of all constraints except one are bounded by a polynomial of the input size. We furthermore derive truthful in expectation mechanisms on general networks with bounded degree.

A genetic algorithm based approach to solve multi-resource multi-objective knapsack problem for vegetable wholesalers in fuzzy environment

Vegetable wholesaling problem has a vital role in the business system. In this problem, a vegetable wholesaler is supposed to supply raw, fresh vegetables to supermarkets in an efficient way by minimizing time but maximizing profit. In this paper, we have presented a multi-resource multi-objective knapsack problem (MRKP) for vegetable wholesalers. This model is beneficial for a vegetable wholesaler who collects different types of vegetables (objects) from different villages (resources) to a market for selling the vegetables (objects). MRKP is an extension of the classical concept of 0–1 multi-dimensional knapsack problem (KP). In this model, precisely a wholesaler has a limited capacity van/trolley by which he/she collects a set of vegetables from different villages/vegetable fields. In this model, we have assumed that all the vegetables are available for each resource. Each type of vegetable is associated with a weight, a corresponding profit, and a collection time (for a particular resource). The profit, weight, and collection time of objects is different for different resources. Here a time slice is considered for each object to collect it from different villages to a particular destination/market. Also, the profit and time are the two objectives. MRKP aims to find the amount of an object and the corresponding resource name from which it is collected. We have solved the proposed problem in the fuzzy environment. In this paper, we have explained two defuzzification techniques, namely fuzzy expectation and total $$\lambda$$ -integral value method to solve the proposed problem. We have explained a modified multi-objective genetic algorithm (NSGA-II by Deb et al. in IEEE Trans Evol Comput 6(2):192–197, 2002) that is to maximise the profit and minimise the time to collect the objects. We have considered a multi-objective benchmark test function to show the effectiveness of the proposed Genetic Algorithm. Modification is made by introducing refinement operation. An extensive computational experimentation has been executed that generates interesting results to establish the effectiveness of the proposed model.

An experimental evaluation of a parallel simulated annealing approach for the 0–1 multidimensional knapsack problem

The recent progresses in the development and improvement of multicore/manycore architectures instigate the design of algorithms capable of exploring the functionalities provided by these architectures to solve more efficiently hard problems. The NP-hard class of problems contains several problems which demand for efficient alternative solutions, since there is a wide range or real-world problems which can be modeled as one of them and it is unknown if they can be exactly solved in feasible time. The 0–1 multidimensional knapsack problem (0–1 MKP) is one of the most emblematic NP-hard problems and this work focuses on the proposal of a parallel simulated annealing algorithm using GPGPU. The results achieved by the parallel SA were compared to other reference works and showed that GPGPU is effective on the task of obtaining better quality solutions in reduced execution time when compared to sequential programs. Our proposed approach can be adapted to other parallel platforms.

Efficient jobs scheduling approach for big data applications

Abstract The MapReduce framework has become a leading scheme for processing large-scale data applications in recent years. However, big data applications executed on computer clusters require a large amount of energy, which costs a considerable fraction of the data center’s overall costs. Therefore, for a data center, how to reduce the energy consumption becomes a critical issue. Although Hadoop YARN adopts fine-grained resource management schemes for job scheduling, it doesn’t consider the energy saving problem. In this paper, an Energy-aware Fair Scheduling framework based on YARN (denoted as EFS) is proposed, which can effectively reduce energy consumption while meet the required Service Level Agreements (SLAs). EFS not only can schedule jobs to energy-efficiency nodes, but also can power on or off the nodes. To do so, the energy-aware dynamic capacity management with deadline-driven policy is used to allocate the resources for MapReduce tasks in terms of the average execution time of containers and users request resources. And then, Energy-aware fair based scheduling problem is modeled as multi-dimensional knapsack problem (MKP) and the energy-aware greedy algorithm (EAGA) is proposed to realize tasks fine-grained placement on energy-efficient nodes. Finally, the nodes which have been kept in idle state for the threshold duration are turned off to reduce energy costs. We perform extensive experiments on the Hadoop YARN clusters to compare the energy consumption and executing time of EFS with some state-of-the-art policies. The experimental results show that EFS can not only keep the proper number of nodes in on states to meet the computing requirements but also achieve the goal of energy savings.

A two-phase tabu-evolutionary algorithm for the 0–1 multidimensional knapsack problem

The 0–1 multidimensional knapsack problem is a well-known NP-hard combinatorial optimization problem with numerous applications. In this work, we present an effective two-phase tabu-evolutionary algorithm for solving this computationally challenging problem. The proposed algorithm integrates two solution-based tabu search methods into the evolutionary framework that applies a hyperplane-constrained crossover operator to generate offspring solutions, a dynamic method to determine search zones of interest, and a diversity-based population updating rule to maintain a healthy population. We show the competitiveness of the proposed algorithm by presenting computational results on the 281 benchmark instances commonly used in the literature. In particular, in a computational comparison with the best algorithms in the literature on multiple data sets, we show that our method on average matches more than twice the number of best known solutions to the harder problems than any other method and in addition yields improved best solutions (new lower bounds) for 4 difficult instances. We investigate two key ingredients of the algorithm to understand their impact on the performance of the algorithm.

The 0/1 Multidimensional Knapsack Problem and Its Variants: A Survey of Practical Models and Heuristic Approaches

The 0/1 Multidimensional Knapsack Problem (0/1 MKP) is an interesting NP-hard combinatorial optimization problem that can model a number of challenging applications in logistics, finance, telecommunications and other fields. In the 0/1 MKP, a set of items is given, each with a size and value, which has to be placed into a knapsack that has a certain number of dimensions having each a limited capacity. The goal is to find a subset of items leading to the maximum total profit while respecting the capacity constraints. Even though the 0/1 MKP is well studied in the literature, we can just find a little number of recent review papers on this problem. Furthermore, the existing reviews focus particularly on some specific issues. This paper aims to give a general and comprehensive survey of the considered problem so that it can be useful for both researchers and practitioners. Indeed, we first describe the 0/1 MKP and its relevant variants. Then, we present the detailed models of some important real-world applications of this problem. Moreover, an important collection of recently published heuristics and metaheuristics is categorized and briefly reviewed. These approaches are then quantitatively compared through some indicative statistics. Finally, some synthetic remarks and research directions are highlighted in the conclusion.

A Class of Continuous Separable Nonlinear Multidimensional Knapsack Problems

The nonlinear multidimensional knapsack problem is defined as the minimization of a convex function with multiple linear constraints. The methods developed for nonlinear multidimensional programming problems are often applied to solve the nonlinear multidimensional knapsack problems, but they are inefficient or limited since most of them do not exploit the characteristics of the knapsack problems. In this paper, by establishing structural properties of the continuous separable nonlinear multidimensional knapsack problem, we develop a multi-tier binary solution method for solving the continuous nonlinear multidimensional knapsack problems with general structure. The computational complexity is polynomial in the number of variables. We presented two examples to illustrate the general application of our method and we used statistical results to show the effectiveness of our method.

Adaptation of cheapest shop seeker algorithm for multidimensional knapsack problem

Adaptation of cheapest shop seeker algorithm for multidimensional knapsack problem

Three pseudo-utility ratio-inspired particle swarm optimization with local search for multidimensional knapsack problem

Abstract In this study, a three-ratio self-adaptive check and repair operator-inspired particle swarm optimization (3R-SACRO-PSO) with neighborhood local search is developed to solve the multidimensional knapsack problem (MKP). The proposed 3R-SACRO-PSO systematically alters substitute pseudo-utility ratios as the PSO method is executed. In addition, a local search scheme is introduced to improve solution quality. The proposed 3R-SACRO-PSO algorithm is tested using 168 different widely used benchmarks from the OR-Library to demonstrate and validate its performance. The control parameters for the performance test are determined through the Taguchi method. Experimental results parallel those of other PSO algorithms, and statistical test results show that the quality and efficiency of the proposed 3R-SACRO are better than those of the two-ratio SACRO method. Moreover, the proposed 3R-SACRO-PSO is on par with state-of-the-art PSO approaches. Thus, introducing the third pseudo-utility ratio into SACRO improves the performance of SACRO-based PSO. The neighborhood local search scheme further improves the solution quality in handling MKPs.