• Journal of Korean Society of Industrial and Systems Engineering
• /
• v.38 no.4
• /
• pp.64-71
• /
• 2015

In this paper, we present a multi-period 0-1 knapsack problem which has the cardinality constraints. Theoretically, the presented problem can be regarded as an extension of the multi-period 0-1 knapsack problem. In the multi-period 0-1 knapsack problem, there are n jobs to be performed during m periods. Each job has the execution time and its completion gives profit. All the n jobs are partitioned into m periods, and the jobs belong to i-th period may be performed not later than in the i-th period, i = 1, ${\cdots}$, m. The total production time for periods from 1 to i is given by $b_i$ for each i = 1, ${\cdots}$, m, and the objective is to maximize the total profit. In the extended problem, we can select a specified number of jobs from each of periods associated with the corresponding cardinality constraints. As the extended problem is NP-hard, the branch and bound method is preferable to solve it, and therefore it is important to have efficient procedures for solving its linear programming relaxed problem. So we intensively explore the LP relaxed problem and suggest a polynomial time algorithm. We first decompose the LP relaxed problem into m subproblems associated with each cardinality constraints. Then we identify some new properties based on the parametric analysis. Finally by exploiting the special structure of the LP relaxed problem, we develop an efficient algorithm for the LP relaxed problem. The developed algorithm has a worst case computational complexity of order max[$O(n^2logn)$, $O(mn^2)$] where m is the number of periods and n is the total number of jobs. We illustrate a numerical example.

• Journal of the military operations research society of Korea
• /
• v.10 no.1
• /
• pp.57-63
• /
• 1984

The 0-1 multi-period Knapsack problem (MPKP) has a horizon of m periods, each having a number of types of projects with values and weights. Subject to the requirement, the cummulative capacity of the problem in each period i cannot be exceeded by the total weight of the projects selected in period 1, 2, ..., i. It is a problem of selecting the projects in such a way that the total value in the knapsack through the horizon of m periods is maximized. A search algorithm is developed and tested in this paper. Search rules that avoid the search of redundant partial solutions are used in the algorithm. Using the property of MPKP, a surrogate constraint concerned with the most available requirement is used in the bounding technique.

• Journal of the Korea Society of Computer and Information
• /
• v.23 no.12
• /
• pp.1-9
• /
• 2018

The multiple-choice multidimensional knapsack problem (MMKP) is a variant of the well known 0-1 knapsack problem, which is known as an NP-hard problem. This paper proposes a method for solving the MMKP using the integer programming-based local search (IPbLS). IPbLS is a kind of a local search and uses integer programming to generate a neighbor solution. The most important thing in IPbLS is the way to select items participating in the next integer programming step. In this paper, three ways to select items are introduced and compared on 37 well-known benchmark data instances. Experimental results shows that the method using linear programming is the best for the MMKP. It also shows that the proposed method can find the equal or better solutions than the best known solutions in 23 data instances, and the new better solutions in 13 instances.

• Management Science and Financial Engineering
• /
• v.10 no.1
• /
• pp.97-106
• /
• 2004

In this paper, we propose an effective cut generation method based on the Chvatal-Gomory procedure for a variable-capacity (0,l)-Knapsack problem with two general integer variables. We first derive a class of valid inequalities for the problem using Chvatal-Gomory procedure, then analyze the associated separation problem. Based on the results, we show that there exists a pseudo-polynomial time algorithm to solve the separation problem. By analyzing the theoretical strength of the inequalities which can be generated by the proposed cut generation method, we show that generated inequalties define facets under mild conditions. We also extend the result to the case in which a nontrivial upper bound is imposed on a general integer variable.

• International Journal of Computer Science & Network Security
• /
• v.24 no.2
• /
• pp.15-24
• /
• 2024

One of the most significant issues in combinatorial optimization is the classical NP-complete conundrum known as the 0/1 Knapsack Problem. This study delves deeply into the investigation of practical solutions, emphasizing two classic algorithmic paradigms, brute force, and dynamic programming, along with the metaheuristic and nature-inspired family algorithm known as the Genetic Algorithm (GA). The research begins with a thorough analysis of the dynamic programming technique, utilizing its ability to handle overlapping subproblems and an ideal substructure. We evaluate the benefits of dynamic programming in the context of the 0/1 Knapsack Problem by carefully dissecting its nuances in contrast to GA. Simultaneously, the study examines the brute force algorithm, a simple yet comprehensive method compared to Branch & Bound. This strategy entails investigating every potential combination, offering a starting point for comparison with more advanced techniques. The paper explores the computational complexity of the brute force approach, highlighting its limitations and usefulness in resolving the 0/1 Knapsack Problem in contrast to the set above of algorithms.

• Journal of the Korean Institute of Intelligent Systems
• /
• v.15 no.6
• /
• pp.730-735
• /
• 2005

Though Knapsack Problem appears to be simple, it is a NP-hard problem that is not solved in polynomial time as combinational optimization problems. To solve this problem, GA(Genetic Algorithms) was used in the past. However, there were difficulties in real experiments because the conventional method didn't reflect the precise characteristics of DNA. In this paper we proposed ACO (Algorithm for Code Optimization) that applies DNA coding method to DNA computing to solve problems of Knapsack Problem. ACO was applied to (0,1) Knapsack Problem; as a result, it reduced experimental errors as compared with conventional methods, and found accurate solutions more rapidly.

• Proceedings of the Korean Operations and Management Science Society Conference
• /
• 2003.11a
• /
• pp.93-96
• /
• 2003

The generalized knapsack problem, or gknap is the combinatorial optimization problem of optimizing a nonnegative linear functional over the integral hull of the intersection of a polynomially separable 0 - 1 polytope and a knapsack constraint. Among many potential applications, the knapsack, the restricted shortest path, and the restricted spanning tree problem are such examples. We establish some necessary and sufficient conditions for a gknap to admit a fully polynomial approximation scheme, or FPTAS, To do so, we recapture the scaling and approximate binary search techniques in the framework of gknap. This also enables us to find a condition that a gknap does not have an FP-TAS. This condition is more general than the strong NP-hardness.

• Journal of Korean Society of Industrial and Systems Engineering
• /
• v.7 no.10
• /
• pp.29-33
• /
• 1984

Many methods have been developed to get a good Computation steps. I think that almost methods of them have been solved by using a theory of [Vj]. But I have thought that it Can be solved by an other method. This method is a way to get a Computations steps by using [Aj] instead of [Vj]. It requires less Computation time than [Vj]. So I think that method is an efficient Algorithm about "the Development of Algorithm method for the 0 - 1 Knapsack problem."

• Journal of the Korean Operations Research and Management Science Society
• /
• v.25 no.3
• /
• pp.1-15
• /
• 2000

In this paper, we propose a practical cut generation method based on the Chvatal-Gomory procedure for the (0, 1)-Knapsack problem with a variable capacity. For a given set N of n items each of which has a positive integral weight and a facility of positive integral capacity, a feasible solution of the problem is defined as a subset S of N along with the number of facilities that can satisfy the sum of weights of all the items in S. We first derive a class of valid inequalities for the problem using Chvatal-Gomory procedure, then analyze the associated separation problem. Based on the results, we develop an affective cut generation method. We then analyze the theoretical strength of the inequalities which can be generated by the proposed cut generation method. Preliminary computational results are also presented which show the effectiveness of the proposed cut generation method.

• Journal of the Korean Operations Research and Management Science Society
• /
• v.17 no.1
• /
• pp.31-41
• /
• 1992

An extension of generalized linear multiple choice knapsack problem [1] is presented and an algorithm of order 0([n .n$_{max}$]$_{2}$) is developed by exploiting its extended properties, where n and n$_{max}$ denote the total number of variables and the cardinality of the largest multiple choice set, respectively. A numerical example is presented and computational aspects are discussed.sed.