Gompertz PSO variants for Knapsack and Multi-Knapsack Problems

Particle Swarm Optimization,a potential swarm intelligence heuristic,has been recognized as a global optimizer for solving various continuous as well as discrete optimization problems.Encourged by the performance of Gompertz PSO on a set of continuous problems,this works extends the application of Gompertz PSO for solving binary optimization problems.Moreover,a new chaotic variant of Gompertz PSO namely Chaotic Gompertz Binary Particle Swarm Optimization(CGBPSO)has also been proposed.The new variant is further analysed for solving binary optimization problems.The new chaotic variant embeds the notion of chaos into GBPSO in later stages of searching process to avoid stagnation phenomena.The efficiency of both the Binary PSO variants has been tested on different sets of Knapsack Problems(KPs):0-1 Knapsack Problem(0-1 KP)and Multidimensional Knapsack Problems(MKP).The concluding remarks have made on the basis of detailed analysis of results,which comprises the comparison of results for Knapsack and Multidimensional Knapsack problems obtained using BPSO,GBPSO and CGBPSO.

SOME EXTENDED KNAPSACK PROBLEMS INVOLVING JOB PARTITION BETWEEN TWO PARTIES

Some novel applications and pragmatic variations of knapsack problem （KP） are presented and constructed, which are formulated and developed from a model initiated in this paper on profit allocation from partition of jobs in terms of two-person discrete cooperation game.

A branch-and-bound algorithm for multi-dimensional quadratic 0-1 knapsack problems

In this paper, a branch-and-bound method for solving multi-dimensional quadratic 0-1 knapsack problems was studied. The method was based on the Lagrangian relaxation and the surrogate constraint technique for finding feasible solutions. The Lagrangian relaxations were solved with the maximum-flow algorithm and the Lagrangian bounds was determined with the outer approximation method. Computational results show the efficiency of the proposed method for multi-dimensional quadratic 0-1 knapsack problems.

Quantum-inspired ant algorithm for knapsack problems

The knapsack problem is a well-known combinatorial optimization problem which has been proved to be NP-hard. This paper proposes a new algorithm called quantum-inspired ant algorithm （QAA） to solve the knapsack problem. QAA takes the advantage of the principles in quantum computing, such as qubit, quantum gate, and quantum superposition of states, to get more probabilistic-based status with small colonies. By updating the pheromone in the ant algorithm and rotating the quantum gate, the algorithm can finally reach the optimal solution. The detailed steps to use QAA are presented, and by solving series of test cases of classical knapsack problems, the effectiveness and generality of the new algorithm are validated.

An efficient algorithm for multi-dimensional nonlinear knapsack problems

Multi-dimensional nonlinear knapsack problem is a bounded nonlinear integer programming problem that maximizes a separable nondecreasing function subject to multiple separable nondecreasing constraints. This problem is often encountered in resource allocation, industrial planning and computer network. In this paper, a new convergent Lagrangian dual method was proposed for solving this problem. Cutting plane method was used to solve the dual problem and to compute the Lagrangian bounds of the primal problem. In order to eliminate the duality gap and thus to guarantee the convergence of the algorithm, domain cut technique was employed to remove certain integer boxes and partition the revised domain to a union of integer boxes. Extensive computational results show that the proposed method is efficient for solving large-scale multi-dimensional nonlinear knapsack problems. Our numerical results also indicate that the cutting plane method significantly outperforms the subgradient method as a dual search procedure.

A Weight-Coded Evolutionary Algorithm for the Multidimensional Knapsack Problem

A revised weight-coded evolutionary algorithm (RWCEA) is proposed for solving multidimensional knapsack problems. This RWCEA uses a new decoding method and incorporates a heuristic method in initialization. Computational results show that the RWCEA performs better than a weight-coded evolutionary algorithm pro-posed by Raidl (1999) and to some existing benchmarks, it can yield better results than the ones reported in the OR-library.

SEMI-DEFINITE RELAXATION ALGORITHM OF MULTIPLE KNAPSACK PROBLEM

The multiple knapsack problem denoted by MKP (B,S,m,n) can be defined as fol- lows.A set B of n items and a set Sof m knapsacks are given such thateach item j has a profit pjand weightwj,and each knapsack i has a capacity Ci.The goal is to find a subset of items of maximum profit such that they have a feasible packing in the knapsacks.MKP(B,S,m,n) is strongly NP- Complete and no polynomial- time approximation algorithm can have an approxima- tion ratio better than0 .5 .In the last ten years,semi- definite programming has been empolyed to solve some combinatorial problems successfully.This paper firstly presents a semi- definite re- laxation algorithm (MKPS) for MKP (B,S,m,n) .It is proved that MKPS have a approxima- tion ratio better than 0 .5 for a subclass of MKP (B,S,m,n) with n≤ 1 0 0 ,m≤ 5 and maxnj=1{ wj} minmi=1{ Ci} ≤ 2 3 .

Uncertain bilevel knapsack problem and its solution

This paper aims at providing an uncertain bilevel knapsack problem (UBKP) model, which is a type of BKPs involving uncertain variables. And then an uncertain solution for the UBKP is proposed by defining PE Nash equilibrium and PE Stackelberg Nash equilibrium. In order to improve the computational efficiency of the uncertain solution, several operators (binary coding distance, inversion operator, explosion operator and binary back learning operator) are applied to the basic fireworks algorithm to design the binary backward fireworks algorithm (BBFWA), which has a good performance in solving the BKP. As an illustration, a case study of the UBKP model and the P-E uncertain solution is applied to an armaments transportation problem.

Surrogate dual method for multi-dimensional nonlinear knapsack problems

Multi-dimensional nonlinear knapsack problems are often encountered in resource allocation, industrial planning and computer networks. In this paper, a surrogate dual method was proposed for solving this class of problems. Multiply constrained problem was relaxed to a singly constrained problem by using the surrogate technique. To compute tighter bounds of the primal problem, the cutting plane method was used to solve the surrogate dual problem, where the surrogate relaxation problem was solved by the 0-1 linearization method. The domain cut technique was employed to eliminate the duality gap and thus to guarantee the convergence of tile algorithm. Numerical results were reported for large-scale multi-dimensional nonlinear knapsack problems.

Improved Parallel Three-List Algorithm for the Knapsack Problem without Memory Conflicts

Based on the two-list algorithm and the parallel three-list algorithm, an improved parallel three-list algorithm for knapsack problem is proposed, in which the method of divide and conquer, and parallel merging without memory conflicts are adopted. To find a solution for the n-element knapsack problem, the proposed algorithm needs O（2^3n/8） time when O（2^3n/8） shared memory units and O（2^n/4） processors are available. The comparisons between the proposed algorithm and 10 existing algorithms show that the improved parallel three-fist algorithm is the first exclusive-read exclusive-write （EREW） parallel algorithm that can solve the knapsack instances in less than O（2^n/2） time when the available hardware resource is smaller than O（2^n/2） , and hence is an improved result over the past researches.

An Optimal Parallel Algorithm for the Knapsack Problem Based on EREW

A new parallel algorithm is proposed for the knapsack problem where the method of divide and conquer is adopted. Based on an EREW-SIMD machine with shared memory, the proposed algorithm utilizes O(2 n/4 ) 1-ε processors, 0≤ ε ≤1, and O(2 n/2 ) memory to find a solution for the n -element knapsack problem in time O(2 n/4 (2 n/4 ) ε) . The cost of the proposed parallel algorithm is O(2 n/2 ) , which is an optimal method for solving the knapsack problem without memory conflicts and an improved result over the past researches.

A Hybrid Parallel Multi-Objective Genetic Algorithm for 0/1 Knapsack Problem

In this paper a hybrid parallel multi-objective genetic algorithm is proposed for solving 0/1 knapsack problem. Multi-objective problems with non-convex and discrete Pareto front can take enormous computation time to converge to the true Pareto front. Hence, the classical multi-objective genetic algorithms (MOGAs) (i.e., non- Parallel MOGAs) may fail to solve such intractable problem in a reasonable amount of time. The proposed hybrid model will combine the best attribute of island and Jakobovic master slave models. We conduct an extensive experimental study in a multi-core system by varying the different size of processors and the result is compared with basic parallel model i.e., master-slave model which is used to parallelize NSGA-II. The experimental results confirm that the hybrid model is showing a clear edge over master-slave model in terms of processing time and approximation to the true Pareto front.

On Merging Cover Inequalities for Multiple Knapsack Problems

This paper describes methods to merge two cover inequalities and also simultaneously merge multiple cover inequalities in a multiple knapsack instance. Theoretical results provide conditions under which merged cover inequalities are valid. Polynomial time algorithms are created to find merged cover inequalities. A computational study demonstrates that merged inequalities improve the solution times for benchmark multiple knapsack instances by about 9% on average over CPLEX with default settings.

Big Data Flow Adjustment Using Knapsack Problem

The advancements of mobile devices, public networks and the Internet of creature huge amounts of complex data, both construct & unstructured are being captured in trust to allow organizations to produce better business decisions as data is now pivotal for an organizations success. These enormous amounts of data are referred to as Big Data, which enables a competitive advantage over rivals when processed and analyzed appropriately. However Big Data Analytics has a few concerns including Management of Data, Privacy & Security, getting optimal path for transport data, and Data Representation. However, the structure of network does not completely match transportation demand, i.e., there still exist a few bottlenecks in the network. This paper presents a new approach to get the optimal path of valuable data movement through a given network based on the knapsack problem. This paper will give value for each piece of data, it depends on the importance of this data (each piece of data defined by two arguments size and value), and the approach tries to find the optimal path from source to destination, a mathematical models are developed to adjust data flows between their shortest paths based on the 0 - 1 knapsack problem. We also take out computational experience using the commercial software Gurobi and a greedy algorithm (GA), respectively. The outcome indicates that the suggest models are active and workable. This paper introduced two different algorithms to study the shortest path problems: the first algorithm studies the shortest path problems when stochastic activates and activities does not depend on weights. The second algorithm studies the shortest path problems depends on weights.

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 Novel Method for Solving Unbounded Knapsack Problem

Knapsack problem is one kind of NP-Complete problem. Unbounded knapsack problems are more complex and harder than general knapsack problem. In this paper,we apply QGAs (Quantum Genetic Algorithms) to solve unbounded knapsack problem and then follow other procedures. First,present the problem into the mode of QGAs and figure out the corresponding genes types and their fitness functions. Then,find the perfect combination of limitation and largest benefit. Finally,the best solution will be found. Primary experiment indicates that our method has well results.

Optimal Parallel Algorithm for the Knapsack Problem Without Memory Conflicts

The knapsack problem is well known to be NP-complete. Due to its importance in cryptosystem and in number theory, in the past two decades, much effort has been made in order to find techniques that could lead to practical algorithms with reasonable running time. This paper proposes a new parallel algorithm for the knapsack problem where the optimal merging algorithm is adopted. The proposed algorithm is based on anEREW-SIMD machine with shared memory. It is proved that the proposed algorithm is both optimal and the first without memory conflicts algorithm for the knapsack problem. The comparisons of algorithm performance show that it is an improvement over the past researches. Keywords knapsack problem - NP-complete - parallel algorithm - optimal algorithm - memory conflict Supported by the National Natural Science Foundation of China under Grant No.60273075, the National High Technology Development 863 Program of China under Grant No.863-306-ZD-11-01-06.Ken-Li Li received his B.S. and M.S. degrees in mathematics from National University of Defense Technology and Central South University in 1995 and 2000 respectively and he is now a Ph.D. candidate in computer software and theory at Huazhong University of Science and Technology. His main research interests include parallel computing and combinatorial optimization.Ren-Fa Li received his Ph.D. degree in computer software and theory at Huazhong University of Science and Technology, and he is concurrently a professor and Ph.D. supervisor in School of Computer and Communication, Human University. His main research interests include network computing.Qing-Hua Li received his M.S. degree in computer science from Huazhong University of Science and Technology in 1981, and he is concurrently a professor and Ph.D. supervisor in School of Computer Science and Technology, Huazhong University of Science and Technology. His current research interests include parallel processing, combinatorial optimization, and grid computing.

1-Neighbour knapsack problem and prospective greedy algorithm of intentional islanding in active distribution network

A connected and undirected graph model of active distribution networks with considering the function of interconnecting switches is constructed in this paper.Based on this model,the island partition problem of active distribution networks can be described as a 1-neighbour knapsack problem.An effective heuristic algorithm named prospective greedy algorithm is then proposed to solve this problem.Case studies on PG&E 69-bus network show the validity of the proposed model and algorithm.

Uncertain bilevel knapsack problem based on an improved binary wolf pack algorithm

To address indeterminism in the bilevel knapsack problem,an uncertain bilevel knapsack problem(UBKP)model is proposed.Then,an uncertain solution for UBKP is proposed by defining thePE Nash equilibrium andPE Stackelberg-Nash equilibrium.To improve the computational efficiency of the uncertain solution,an evolutionary algorithm,the improved binary wolf pack algorithm,is constructed with one rule(wolf leader regulation),two operators(invert operator and move operator),and three intelligent behaviors(scouting behavior,intelligent hunting behavior,and upgrading).The UBKP model and thePE uncertain solution are applied to an armament transportation problem as a case study.

The Continuous Knapsack Problem with Capacities

We address a variant of the continuous knapsack problem,where capacities regarding costs of items are given into account.We prove that the problem is NP-complete although the classical continuous knapsack problem is solvable in linear time.For the case that there exists exactly one capacity for all items,we solve the corresponding problem in O(n log n)time,where n is the number of items.