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 G...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.展开更多
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...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.展开更多
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 ...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.展开更多
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 prob...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.展开更多
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 i...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.展开更多
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 prob...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.展开更多
The knapsack problem is a classical combinatorial optimization problem widely encountered in areas such as logistics,resource allocation,and portfolio optimization.Traditional methods,including dynamic program-ming(DP...The knapsack problem is a classical combinatorial optimization problem widely encountered in areas such as logistics,resource allocation,and portfolio optimization.Traditional methods,including dynamic program-ming(DP)and greedy algorithms,have been effective in solving small problem instances but often struggle with scalability and efficiency as the problem size increases.DP,for instance,has exponential time complexity and can become computationally prohibitive for large problem instances.On the other hand,greedy algorithms offer faster solutions but may not always yield the optimal results,especially when the problem involves complex constraints or large numbers of items.This paper introduces a novel reinforcement learning(RL)approach to solve the knapsack problem by enhancing the state representation within the learning environment.We propose a representation where item weights and volumes are expressed as ratios relative to the knapsack’s capacity,and item values are normalized to represent their percentage of the total value across all items.This novel state modification leads to a 5%improvement in accuracy compared to the state-of-the-art RL-based algorithms,while significantly reducing execution time.Our RL-based method outperforms DP by over 9000 times in terms of speed,making it highly scalable for larger problem instances.Furthermore,we improve the performance of the RL model by incorporating Noisy layers into the neural network architecture.The addition of Noisy layers enhances the exploration capabilities of the agent,resulting in an additional accuracy boost of 0.2%–0.5%.The results demonstrate that our approach not only outperforms existing RL techniques,such as the Transformer model in terms of accuracy,but also provides a substantial improvement than DP in computational efficiency.This combination of enhanced accuracy and speed presents a promising solution for tackling large-scale optimization problems in real-world applications,where both precision and time are critical factors.展开更多
The multiple knapsack problem （MKP） forms a base for resolving many real-life problems. This has also been considered with multiple objectives in genetic algorithms （GAs） for proving its efficiency. GAs use self- ...The multiple knapsack problem （MKP） forms a base for resolving many real-life problems. This has also been considered with multiple objectives in genetic algorithms （GAs） for proving its efficiency. GAs use self- adaptability to effectively solve complex problems with constraints, but in certain cases, self-adaptability fails by converging toward an infeasible region. This pitfall can be resolved by using different existing repairing techniques; however, this cannot assure convergence toward attaining the optimal solution. To overcome this issue, gene position-based suppression （GPS） has been modeled and embedded as a new phase in a classical GA. This phase works on the genes of a newly generated individual after the recombination phase to retain the solution vector within its feasible region and to im- prove the solution vector to attain the optimal solution. Genes holding the highest expressibility are reserved into a subset, as the best genes identified from the current individuals by re- placing the weaker genes from the subset. This subset is used by the next generated individual to improve the solution vec- tor and to retain the best genes of the individuals. Each gene＇s positional point and its genotype exposure for each region in an environment are used to fit the best unique genes. Further, suppression of expression in conflicting gene＇s relies on the requirement toward the level of exposure in the environment or in eliminating the duplicate genes from the environment.The MKP benchmark instances from the OR-library are taken for the experiment to test the new model. The outcome por- trays that GPS in a classical GA is superior in most of the cases compared to the other existing repairing techniques.展开更多
in this paper, the approximation for four kinds of knapsack prob- lems with multiple constraints is studied: 0/1 Multiple Constraint Knapsack Problem(0/1 MCKP), Integer Multiple Constraint Knapsack Problem (Integer MC...in this paper, the approximation for four kinds of knapsack prob- lems with multiple constraints is studied: 0/1 Multiple Constraint Knapsack Problem(0/1 MCKP), Integer Multiple Constraint Knapsack Problem (Integer MCKP), 0/1k-Constraillt Knapsack Problem (0/1 k-CKP) and Integer k-Constraint KnapsackProblem (Integer k-CKP). The following results are obtained:1) Unless NP = co - R, no polynomial time algorithm approximates 0/1 MCKPor Integer MCKP within a factor k(1/2)- for any > 0; unless NP = P, nopolynomial time algorithm approximates 0/1 MCKP or integer MCKP within afactor k(1/4)- for any > 0, where k stands for the number of constraints.2) For any fixed positive integer k, 0/1 k-CKP has a fully polynomial time approximation scheme (FPTAS).3) For any fixed positive integer k, Integer k-CKP has a fast FPTAS which hastime complexity O(n +) and space complexity O(n + (1/3)), andfinds an approximate solution to within 5 of the optimal solution.展开更多
Membrane algorithms are a class of distributed and parallel algorithms inspired by the structure and behavior of living cells. Many attractive features of living cells have already been abstracted as operators to impr...Membrane algorithms are a class of distributed and parallel algorithms inspired by the structure and behavior of living cells. Many attractive features of living cells have already been abstracted as operators to improve the performance of algorithms. In this work, inspired by the function of biological neuron cells storing information, we consider a memory mechanism by introducing memory modules into a membrane algorithm. The framework of the algorithm consists of two kinds of modules (computation modules and memory modules), both of which are arranged in a ring neighborhood topology. They can store and process information, and exchange information with each other. We test our method on a knapsack problem to demonstrate its feasibility and effectiveness. During the process of approaching the optimum solution, feasible solutions are evolved by rewriting rules in each module, and the information transfers according to directions defined by communication rules. Simulation results showed that the performance of membrane algorithms with memory cells is superior to that of algorithms without memory cells for solving a knapsack problem. Furthermore, the memory mechanism can prevent premature convergence and increase the possibility of finding a global solution.展开更多
This paper focuses on the 2-median location improvement problem on tree networks and the problem is to modify the weights of edges at the minimum cost such that the overall sum of the weighted distance of the vertices...This paper focuses on the 2-median location improvement problem on tree networks and the problem is to modify the weights of edges at the minimum cost such that the overall sum of the weighted distance of the vertices to the respective closest one of two prescribed vertices in the modified network is upper bounded by a given value.l1 norm and l∞norm are used to measure the total modification cost. These two problems have a strong practical application background and important theoretical research value. It is shown that such problems can be transformed into a series of sum-type and bottleneck-type continuous knapsack problems respectively.Based on the property of the optimal solution two O n2 algorithms for solving the two problems are proposed where n is the number of vertices on the tree.展开更多
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. Computatio...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.展开更多
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 ca...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 .展开更多
The set-union knapsack problem(SUKP)is proved to be a strongly NP-hard problem,and it is an extension of the classic NP-hard problem:the 0-1 knapsack problem(KP).Solving the SUKP through exact approaches is computatio...The set-union knapsack problem(SUKP)is proved to be a strongly NP-hard problem,and it is an extension of the classic NP-hard problem:the 0-1 knapsack problem(KP).Solving the SUKP through exact approaches is computationally expensive.Therefore,several swarm intelligent algorithms have been proposed in order to solve the SUKP.Hyper-heuristics have received notable attention by researchers in recent years,and they are successfully applied to solve the combinatorial optimization problems.In this article,we propose a binary particle swarm optimization(BPSO)based hyper-heuristic for solving the SUKP,in which the BPSO is employed as a search methodology.The proposed approach has been evaluated on three sets of SUKP instances.The results are compared with 6 approaches:BABC,EMS,gPSO,DHJaya,b WSA,and HBPSO/TS,and demonstrate that the proposed approach for the SUKP outperforms other approaches.展开更多
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 equil...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.展开更多
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 withou...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.展开更多
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 ...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.展开更多
Reversible data hiding is a confidential communication technique that takes advantage of image file characteristics,which allows us to hide sensitive data in image files.In this paper,we propose a novel high-fidelity ...Reversible data hiding is a confidential communication technique that takes advantage of image file characteristics,which allows us to hide sensitive data in image files.In this paper,we propose a novel high-fidelity reversible data hiding scheme.Based on the advantage of the multipredictor mechanism,we combine two effective prediction schemes to improve prediction accuracy.In addition,the multihistogram technique is utilized to further improve the image quality of the stego image.Moreover,a model of the grouped knapsack problem is used to speed up the search for the suitable embedding bin in each sub-histogram.Experimental results show that the quality of the stego image of our scheme outperforms state-of-the-art schemes in most cases.展开更多
Aiming at the problems of convergence-slow and convergence-free of Discrete Particle Swarm Optimization Algorithm(DPSO) in solving large scale or complicated discrete problem, this article proposes Intuitionistic Fuzz...Aiming at the problems of convergence-slow and convergence-free of Discrete Particle Swarm Optimization Algorithm(DPSO) in solving large scale or complicated discrete problem, this article proposes Intuitionistic Fuzzy Entropy of Discrete Particle Swarm Optimization(IFDPSO) and makes it applied to Dynamic Weapon Target Assignment(WTA). First, the strategy of choosing intuitionistic fuzzy parameters of particle swarm is defined, making intuitionistic fuzzy entropy as a basic parameter for measure and velocity mutation. Second, through analyzing the defects of DPSO, an adjusting parameter for balancing two cognition, velocity mutation mechanism and position mutation strategy are designed, and then two sets of improved and derivative algorithms for IFDPSO are put forward, which ensures the IFDPSO possibly search as much as possible sub-optimal positions and its neighborhood and the algorithm ability of searching global optimal value in solving large scale 0-1 knapsack problem is intensified. Third, focusing on the problem of WTA, some parameters including dynamic parameter for shifting firepower and constraints are designed to solve the problems of weapon target assignment. In addition, WTA Optimization Model with time and resource constraints is finally set up, which also intensifies the algorithm ability of searching global and local best value in the solution of WTA problem. Finally, the superiority of IFDPSO is proved by several simulation experiments. Particularly, IFDPSO, IFDPSO1~IFDPSO3 are respectively effective in solving large scale, medium scale or strict constraint problems such as 0-1 knapsack problem and WTA problem.展开更多
Similar to the classical meet-in-the-middle algorithm,the storage and computation complexity are the key factors that decide the efficiency of the quantum meet-in-the-middle algorithm.Aiming at the target vector of fi...Similar to the classical meet-in-the-middle algorithm,the storage and computation complexity are the key factors that decide the efficiency of the quantum meet-in-the-middle algorithm.Aiming at the target vector of fixed weight,based on the quantum meet-in-the-middle algorithm,the algorithm for searching all n-product vectors with the same weight is presented,whose complexity is better than the exhaustive search algorithm.And the algorithm can reduce the storage complexity of the quantum meet-in-the-middle search algorithm.Then based on the algorithm and the knapsack vector of the Chor-Rivest public-key crypto of fixed weight d,we present a general quantum meet-in-th√e-middle search algorithm based on the target solution of fixed weight,whose computational complexity is∑(d to j=0)(O((1/2)(C^(d-j)_(n-k+1))+O(C^j_klog C^j_k))with∑(d to i=0)C^i_k memory cost.And the optimal value of k is given.Compared to thequantum meet-in-the-middle search algorithm for knapsack problem and the quantum algorithm for searching a target solution of fixed weight,the computational complexity of the algorithm is lower.And its storage complexity is smaller than the quantum meet-in-the-middle-algorithm.展开更多
文摘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.
基金Supported by the Research Fund of Shenzhen University(200552).
文摘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.
基金Project supported by the National Natural Science Foundation of China （Grant No.10571116）
文摘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.
基金supported by the National Natural Science Foundation of China(70871081)the Shanghai Leading Academic Discipline Project(S30504).
文摘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.
文摘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.
基金partially supported by the National Natural Science Foundation of China （Grant Nos.10271073, 10571116）
文摘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.
基金supported in part by the Research Start-Up Funds of South-Central Minzu University under Grants YZZ23002,YZY23001,and YZZ18006in part by the Hubei Provincial Natural Science Foundation of China under Grants 2024AFB842 and 2023AFB202+3 种基金in part by the Knowledge Innovation Program of Wuhan Basic Research underGrant 2023010201010151in part by the Spring Sunshine Program of Ministry of Education of the People’s Republic of China under Grant HZKY20220331in part by the Funds for Academic Innovation Teams and Research Platformof South-CentralMinzu University Grant Number:XT224003,PTZ24001in part by the Career Development Fund(CDF)of the Agency for Science,Technology and Research(A*STAR)(Grant Number:C233312007).
文摘The knapsack problem is a classical combinatorial optimization problem widely encountered in areas such as logistics,resource allocation,and portfolio optimization.Traditional methods,including dynamic program-ming(DP)and greedy algorithms,have been effective in solving small problem instances but often struggle with scalability and efficiency as the problem size increases.DP,for instance,has exponential time complexity and can become computationally prohibitive for large problem instances.On the other hand,greedy algorithms offer faster solutions but may not always yield the optimal results,especially when the problem involves complex constraints or large numbers of items.This paper introduces a novel reinforcement learning(RL)approach to solve the knapsack problem by enhancing the state representation within the learning environment.We propose a representation where item weights and volumes are expressed as ratios relative to the knapsack’s capacity,and item values are normalized to represent their percentage of the total value across all items.This novel state modification leads to a 5%improvement in accuracy compared to the state-of-the-art RL-based algorithms,while significantly reducing execution time.Our RL-based method outperforms DP by over 9000 times in terms of speed,making it highly scalable for larger problem instances.Furthermore,we improve the performance of the RL model by incorporating Noisy layers into the neural network architecture.The addition of Noisy layers enhances the exploration capabilities of the agent,resulting in an additional accuracy boost of 0.2%–0.5%.The results demonstrate that our approach not only outperforms existing RL techniques,such as the Transformer model in terms of accuracy,but also provides a substantial improvement than DP in computational efficiency.This combination of enhanced accuracy and speed presents a promising solution for tackling large-scale optimization problems in real-world applications,where both precision and time are critical factors.
文摘The multiple knapsack problem （MKP） forms a base for resolving many real-life problems. This has also been considered with multiple objectives in genetic algorithms （GAs） for proving its efficiency. GAs use self- adaptability to effectively solve complex problems with constraints, but in certain cases, self-adaptability fails by converging toward an infeasible region. This pitfall can be resolved by using different existing repairing techniques; however, this cannot assure convergence toward attaining the optimal solution. To overcome this issue, gene position-based suppression （GPS） has been modeled and embedded as a new phase in a classical GA. This phase works on the genes of a newly generated individual after the recombination phase to retain the solution vector within its feasible region and to im- prove the solution vector to attain the optimal solution. Genes holding the highest expressibility are reserved into a subset, as the best genes identified from the current individuals by re- placing the weaker genes from the subset. This subset is used by the next generated individual to improve the solution vec- tor and to retain the best genes of the individuals. Each gene＇s positional point and its genotype exposure for each region in an environment are used to fit the best unique genes. Further, suppression of expression in conflicting gene＇s relies on the requirement toward the level of exposure in the environment or in eliminating the duplicate genes from the environment.The MKP benchmark instances from the OR-library are taken for the experiment to test the new model. The outcome por- trays that GPS in a classical GA is superior in most of the cases compared to the other existing repairing techniques.
文摘in this paper, the approximation for four kinds of knapsack prob- lems with multiple constraints is studied: 0/1 Multiple Constraint Knapsack Problem(0/1 MCKP), Integer Multiple Constraint Knapsack Problem (Integer MCKP), 0/1k-Constraillt Knapsack Problem (0/1 k-CKP) and Integer k-Constraint KnapsackProblem (Integer k-CKP). The following results are obtained:1) Unless NP = co - R, no polynomial time algorithm approximates 0/1 MCKPor Integer MCKP within a factor k(1/2)- for any > 0; unless NP = P, nopolynomial time algorithm approximates 0/1 MCKP or integer MCKP within afactor k(1/4)- for any > 0, where k stands for the number of constraints.2) For any fixed positive integer k, 0/1 k-CKP has a fully polynomial time approximation scheme (FPTAS).3) For any fixed positive integer k, Integer k-CKP has a fast FPTAS which hastime complexity O(n +) and space complexity O(n + (1/3)), andfinds an approximate solution to within 5 of the optimal solution.
基金Project supported by the National Natural Science Foundation of China(Nos. 61033003, 91130034, 61100145, 60903105, and 61272071)the PhD Programs Foundation of the Ministry of Education of China(Nos. 20100142110072 and 2012014213008)the Natural Science Foundation of Hubei Province, China (No. 2011CDA027)
文摘Membrane algorithms are a class of distributed and parallel algorithms inspired by the structure and behavior of living cells. Many attractive features of living cells have already been abstracted as operators to improve the performance of algorithms. In this work, inspired by the function of biological neuron cells storing information, we consider a memory mechanism by introducing memory modules into a membrane algorithm. The framework of the algorithm consists of two kinds of modules (computation modules and memory modules), both of which are arranged in a ring neighborhood topology. They can store and process information, and exchange information with each other. We test our method on a knapsack problem to demonstrate its feasibility and effectiveness. During the process of approaching the optimum solution, feasible solutions are evolved by rewriting rules in each module, and the information transfers according to directions defined by communication rules. Simulation results showed that the performance of membrane algorithms with memory cells is superior to that of algorithms without memory cells for solving a knapsack problem. Furthermore, the memory mechanism can prevent premature convergence and increase the possibility of finding a global solution.
基金The National Natural Science Foundation of China(No.10801031)
文摘This paper focuses on the 2-median location improvement problem on tree networks and the problem is to modify the weights of edges at the minimum cost such that the overall sum of the weighted distance of the vertices to the respective closest one of two prescribed vertices in the modified network is upper bounded by a given value.l1 norm and l∞norm are used to measure the total modification cost. These two problems have a strong practical application background and important theoretical research value. It is shown that such problems can be transformed into a series of sum-type and bottleneck-type continuous knapsack problems respectively.Based on the property of the optimal solution two O n2 algorithms for solving the two problems are proposed where n is the number of vertices on the tree.
文摘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.
基金Supported by the National Natural Science Foundation of China(1 9971 0 78)
文摘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 .
基金Supported partly by the Natural Science Foundation of Fujian Province(2020J01843)the Science and Technology Project of the Education Bureau of Fujian(JAT200403)
文摘The set-union knapsack problem(SUKP)is proved to be a strongly NP-hard problem,and it is an extension of the classic NP-hard problem:the 0-1 knapsack problem(KP).Solving the SUKP through exact approaches is computationally expensive.Therefore,several swarm intelligent algorithms have been proposed in order to solve the SUKP.Hyper-heuristics have received notable attention by researchers in recent years,and they are successfully applied to solve the combinatorial optimization problems.In this article,we propose a binary particle swarm optimization(BPSO)based hyper-heuristic for solving the SUKP,in which the BPSO is employed as a search methodology.The proposed approach has been evaluated on three sets of SUKP instances.The results are compared with 6 approaches:BABC,EMS,gPSO,DHJaya,b WSA,and HBPSO/TS,and demonstrate that the proposed approach for the SUKP outperforms other approaches.
基金supported by the National Natural Science Foundation of China(7160118361502522)
文摘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.
文摘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.
文摘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.
基金funded by National Science Council,Taiwan,the Grant Number is NSC 111-2410-H-167-005-MY2.
文摘Reversible data hiding is a confidential communication technique that takes advantage of image file characteristics,which allows us to hide sensitive data in image files.In this paper,we propose a novel high-fidelity reversible data hiding scheme.Based on the advantage of the multipredictor mechanism,we combine two effective prediction schemes to improve prediction accuracy.In addition,the multihistogram technique is utilized to further improve the image quality of the stego image.Moreover,a model of the grouped knapsack problem is used to speed up the search for the suitable embedding bin in each sub-histogram.Experimental results show that the quality of the stego image of our scheme outperforms state-of-the-art schemes in most cases.
基金supported by The National Natural Science Foundation of China under Grant Nos.61402517, 61573375The Foundation of State Key Laboratory of Astronautic Dynamics of China under Grant No. 2016ADL-DW0302+2 种基金The Postdoctoral Science Foundation of China under Grant Nos. 2013M542331, 2015M572778The Natural Science Foundation of Shaanxi Province of China under Grant No. 2013JQ8035The Aviation Science Foundation of China under Grant No. 20151996015
文摘Aiming at the problems of convergence-slow and convergence-free of Discrete Particle Swarm Optimization Algorithm(DPSO) in solving large scale or complicated discrete problem, this article proposes Intuitionistic Fuzzy Entropy of Discrete Particle Swarm Optimization(IFDPSO) and makes it applied to Dynamic Weapon Target Assignment(WTA). First, the strategy of choosing intuitionistic fuzzy parameters of particle swarm is defined, making intuitionistic fuzzy entropy as a basic parameter for measure and velocity mutation. Second, through analyzing the defects of DPSO, an adjusting parameter for balancing two cognition, velocity mutation mechanism and position mutation strategy are designed, and then two sets of improved and derivative algorithms for IFDPSO are put forward, which ensures the IFDPSO possibly search as much as possible sub-optimal positions and its neighborhood and the algorithm ability of searching global optimal value in solving large scale 0-1 knapsack problem is intensified. Third, focusing on the problem of WTA, some parameters including dynamic parameter for shifting firepower and constraints are designed to solve the problems of weapon target assignment. In addition, WTA Optimization Model with time and resource constraints is finally set up, which also intensifies the algorithm ability of searching global and local best value in the solution of WTA problem. Finally, the superiority of IFDPSO is proved by several simulation experiments. Particularly, IFDPSO, IFDPSO1~IFDPSO3 are respectively effective in solving large scale, medium scale or strict constraint problems such as 0-1 knapsack problem and WTA problem.
基金Supported by the National Basic Research Program of China under Grant No.2013CB338002the National Natural Science Foundation of China under Grant No.61502526
文摘Similar to the classical meet-in-the-middle algorithm,the storage and computation complexity are the key factors that decide the efficiency of the quantum meet-in-the-middle algorithm.Aiming at the target vector of fixed weight,based on the quantum meet-in-the-middle algorithm,the algorithm for searching all n-product vectors with the same weight is presented,whose complexity is better than the exhaustive search algorithm.And the algorithm can reduce the storage complexity of the quantum meet-in-the-middle search algorithm.Then based on the algorithm and the knapsack vector of the Chor-Rivest public-key crypto of fixed weight d,we present a general quantum meet-in-th√e-middle search algorithm based on the target solution of fixed weight,whose computational complexity is∑(d to j=0)(O((1/2)(C^(d-j)_(n-k+1))+O(C^j_klog C^j_k))with∑(d to i=0)C^i_k memory cost.And the optimal value of k is given.Compared to thequantum meet-in-the-middle search algorithm for knapsack problem and the quantum algorithm for searching a target solution of fixed weight,the computational complexity of the algorithm is lower.And its storage complexity is smaller than the quantum meet-in-the-middle-algorithm.
