Continuous Knapsack Problem Research Articles

The Genetic Edge: Revolutionizing Multi-Constraint Fractional Knapsack Solutions

Under linear constraints, a greedy algorithm effectively solves the fractional knapsack problem. However, when additional constraints, such as weight and risk, are added, the complexity of the approach rises. Previous studies have shown that the greedy strategy is optimal under single linear constraints and have provided comprehensive documentation of its efficacy in solving the fractional knapsack problem in straightforward, unconstrained circumstances. Nevertheless, there hasn't been much research done on using greedy algorithms to solve issues with many constraints. This study examines how well genetic algorithms (GAs) perform in comparison to the greedy technique for solving the fractional knapsack problem under conditions with various restrictions. Our results show that, although the greedy method is still efficient in linear or unconstrained circumstances, GAs perform better when dealing with various constraints and provide better solutions in spite of their increased complexity of calculation. This study demonstrates the benefits of using evolutionary algorithms to solve difficult restricted optimization issues when more conventional approaches fall short.

The Logic and Application of Greedy Algorithms

Abstract. Being easy to implement and offering faster solutions, the greedy algorithms have widely been used in solving combinatorial optimization problems in computer science and many practical applications, such as resource allocation and data compression. This paper aims at analyzing the definitions, theory, and application of greedy algorithms in computer science as well as their logic and efficacy in numerous situations. In this analysis, concepts such as the greedy choice property and optimal substructure are examined as essential to understanding how greedy algorithms operate. The provided analysis mainly assumes prior knowledge of concepts and problems like the Fractional Knapsack problem and the Shortest Path problem solution as well as some practical problems, for example, resource allocation and the exact coin change problem. Analytical data and real applications are used to analyze the performance of greedy algorithms vis-a-vis other categories, such as the dynamic programming paradigm. The results reveal that greedy algorithms have the potential for great efficiency while also being restricted by certain parameters. They are ideal in the following aspects pertaining to computational aspects, that is, speed, ease of implementation, and applicability to large datasets. But they do not necessarily promise strong optimality in certain situations. The conclusion notes that greedy algorithms deserve further applications when the specified conditions are met and states that the effectiveness of these conditions needs to be studied.

Robust knapsack ordering for a partially-informed newsvendor with budget constraint

This paper studies a robust version of the multi-item newsvendor problem with limited budget. The demand distribution belongs to an ambiguity set that contains all distributions that share the same range, mean and mean absolute deviation. The resulting optimization problem turns out to be solvable by a method reminiscent of the greedy algorithm for continuous knapsack problems, purchasing items in order of marginal effect on the total cost until the budget is spent.

정수 선형 분수 계획법 문제에 대한 파라메트릭 접근법

In this paper, we focus on the integer linear fractional optimization problem, a special case of fractional programming where all functions in the objective function and constraints are linear and all variables are bounded and discrete. To derive the optimal solution, the parametric algorithms are considered, and the efficiency of the algorithms is investigated computationally. Unlike traditional parametric algorithms such as Newton's method, which create a unidirectional sequence approaching the optimal function value, our proposed algorithm generates both upper and lower bounds converging to this value. To demonstrate its effectiveness across various production and operations management problems, the suggested algorithms are used to solve the fractional knapsack problem by comparison to other algorithms (e.g., Newton’s method) under the randomized experimental conditions. The relative practical performance measured by the number of function calls demonstrates that the proposed algorithms are fast and robust for solving the linear fractional programs with discrete variables. Leveraging this algorithm holds the potential to overcome situations in traditional production and operations problems where non-fractional objective functions were previously unconsidered, thereby expecting to derive new outcomes and significance.

The max-sum inverse median location problem on trees with budget constraint

The theory of inverse location involves modifying parameters in such a way that the total cost is minimized and one/several prespecified facilities become optimal based on these perturbed parameters. When the modifying parameters are grouped into sets, with each group's cost measured under the rectilinear norm and the overall cost measured under the Chebyshev norm, the resulting problem is known as the max-sum inverse location problem. This paper addresses the max-sum inverse median location problem on trees with a budget constraint, where the objective is to modify the vertex weights so that a specified vertex becomes a 1-median, while minimizing the max-sum objective within the available budget. To solve this problem, a uni-variable optimization problem is first induced, where the objective function for each specified value of the variable can be obtained through a continuous knapsack problem. Leveraging the monotonicity of the cost function, a combinatorial algorithm is developed, which solves the problem in O(nlog⁡n) time, where n denotes the number of vertices present in the tree.

Optimal targeted mass screening in non‐uniform populations with multiple tests and schemes

AbstractWe study the problem of designing optimal targeted mass screening of non‐uniform populations. Mass screening is an essential tool that is widely utilized in a variety of settings, for example, preventing infertility through screening programs for sexually transmitted diseases, ensuring a safe blood supply for transfusion, and mitigating the transmission of infectious diseases. The objective of mass screening is to maximize the overall classification accuracy under limited budget. In this paper, we address this problem by proposing a proactive optimization‐based framework that factors in population heterogeneity, limited budget, different testing schemes, the availability of multiple assays, and imperfect assays. By analyzing the resulting optimization problem, we take advantage of the structure of the problem as a multi‐dimensional fractional knapsack problem and identify an efficient globally convergent threshold‐style solution scheme that fully characterizes an optimal solution across the entire budget spectrum. Using real‐world data, we conduct a geographic‐based nationwide case study on targeted COVID‐19 screening in the United States. Our results reveal that the identified screening strategies substantially outperform conventional practices by significantly lowering misclassifications while utilizing the same amount of budget. Moreover, our results provide valuable managerial insights with regard to the distribution of testing schemes, assays, and budget across different geographic regions.

Flexible supplier selection and order allocation in the big data era with various quantity discounts.

This paper studies the flexible large-scale supplier selection and order allocation problem with various quantity discounts, i.e., no discount, all-unit discount, incremental discount, and carload discount. It fills a literature gap that models usually formulate one or seldom two types because of the modeling and solution difficulty. All suppliers offering the same discount are far from reality, especially when the number of suppliers is large. The proposed model is a variant of the NP-hard knapsack problem. The greedy algorithm, which solves the fractional knapsack problem optimally, is applied to cope with the challenge. Three greedy algorithms are developed using a problem property and two sorted lists. Simulations show the average optimality gaps are 0.1026%, 0.0547%, and 0.0234% and the model can be solved in centiseconds, densiseconds, and seconds for supplier numbers 1000, 10000, and 100000. This allows the full use of data in the big data era.

Job scheduling for maximum revenue on uniform, parallel machines with major and minor setups and job splitting

We consider a revenue maximization scheduling problem where jobs can be split on parallel, uniform machines. By assuming that only major and minor setups exist between jobs and that these can be grouped into families, we reformulate the scheduling problem as a continuous knapsack problem with setup time. We then create the Revenue Rate Heuristic to solve the maximum revenue scheduling problem for both the partial and binary revenue cases. This makes use of a revenue rate – the ratio between total revenue that can be obtained from a job and the time required to achieve the revenue. The Revenue Rate Heuristic schedules the job with the highest revenue rate first and updates all the revenue rates accordingly based on existing scheduled families and remaining capacity. In cases where there are no setup times or if there is only one machine and an ordering of job revenue and production time, we are able to show that the Revenue Rate Heuristic is optimal. Numerical studies show that the Revenue Rate Heuristic performs well with an average optimality gap of 3.47% across a large set of parameters and when the setup times across families are different from each other. In a case study based on firm-level data, we also show that the average optimality gap remains small.

Cardinality-Constrained Continuous Knapsack Problem with Concave Piecewise-Linear Utilities

Cardinality-Constrained Continuous Knapsack Problem with Concave Piecewise-Linear Utilities

Femtocell deployment for scalable video transmission in 5G networks

In this paper, we propose a femtocell deployment (FD) scheme to transmit scalable video traffic in 5G Heterogeneous Networks (HetNets). Our goal is to use the flexibility of scalable video coding (SVC) for providing video service for the maximum number of users with the best quality of experience (QoE) and the lowest power consumption. The number of required Femto Base Stations (FBSs) is determined by solving a special case of a multiple fractional knapsack problem (MFKP) with three objectives called MU, MQ, and MP. They are defined as maximizing the number of users receiving video service, maximizing the mean QoE, and minimizing power consumption, respectively. To this end, first, a near-optimal solution to the problem is achieved by genetic algorithm (GA). Then, we propose a maximum resource-efficient (ME) solution based on a greedy algorithm which is close to the near-optimal solution with lower computational complexity. The performance evaluation of the network with FD shows that deploying FBSs provides video streaming services for a greater number of users with higher QoE and lower power consumption compared to the network without FD. Moreover, the combination of FD and SVC improves the network performance compared to the combination with non-scalable video bitstreams. Due to the flexibility of scalable video, by using a scalable video bitstream, instead of increasing the number of FBSs, enough capacity for serving more users is provided by decreasing the number of received video layers.

The Stochastic Bilevel Continuous Knapsack Problem with Uncertain Follower’s Objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack, while the follower chooses a feasible packing maximizing his own profit. The leader’s aim is to optimize a linear objective function in the capacity and in the follower’s solution, but with respect to different item values. We address a stochastic version of this problem where the follower’s profits are uncertain from the leader’s perspective, and only a probability distribution is known. Assuming that the leader aims at optimizing the expected value of her objective function, we first observe that the stochastic problem is tractable as long as the possible scenarios are given explicitly as part of the input, which also allows to deal with general distributions using a sample average approximation. For the case of independently and uniformly distributed item values, we show that the problem is #P-hard in general, and the same is true even for evaluating the leader’s objective function. Nevertheless, we present pseudo-polynomial time algorithms for this case, running in time linear in the total size of the items. Based on this, we derive an additive approximation scheme for the general case of independently distributed item values, which runs in pseudo-polynomial time.

The robust bilevel continuous knapsack problem with uncertain coefficients in the follower’s objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack and the follower chooses an optimal packing according to his own profits, which may differ from those of the leader. To this bilevel problem, we add uncertainty in a natural way, assuming that the leader does not have full knowledge about the follower’s problem. More precisely, adopting the robust optimization approach and assuming that the follower’s profits belong to a given uncertainty set, our aim is to compute a solution that optimizes the worst-case follower’s reaction from the leader’s perspective. By investigating the complexity of this problem with respect to different types of uncertainty sets, we make first steps towards better understanding the combination of bilevel optimization and robust combinatorial optimization. We show that the problem can be solved in polynomial time for both discrete and interval uncertainty, but that the same problem becomes NP-hard when each coefficient can independently assume only a finite number of values. In particular, this demonstrates that replacing uncertainty sets by their convex hulls may change the problem significantly, in contrast to the situation in classical single-level robust optimization. For general polytopal uncertainty, the problem again turns out to be NP-hard, and the same is true for ellipsoidal uncertainty even in the uncorrelated case. All presented hardness results already apply to the evaluation of the leader’s objective function.

The sum of root-leaf distance interdiction problem by upgrading edges/nodes on trees.

Network interdiction problems by upgading critical edges/nodes have important applications to reduce the infectivity of the COVID-19. A network of confirmed cases can be described as a rooted tree that has a weight of infectious intensity for each edge. Upgrading edges (nodes) can reduce the infectious intensity with contacts by taking prevention measures such as disinfection (treating the confirmed cases, isolating their close contacts or vaccinating the uninfected people). We take the sum of root-leaf distance on a rooted tree as the whole infectious intensity of the tree. Hence, we consider the sum of root-leaf distance interdiction problem by upgrading edges/nodes on trees (SDIPT-UE/N). The problem (SDIPT-UE) aims to minimize the sum of root-leaf distance by reducing the weights of some critical edges such that the upgrade cost under some measurement is upper-bounded by a given value. Different from the problem (SDIPT-UE), the problem (SDIPT-UN) aims to upgrade a set of critical nodes to reduce the weights of the edges adjacent to the nodes. The relevant minimum cost problem (MCSDIPT-UE/N) aims to minimize the upgrade cost on the premise that the sum of root-leaf distance is upper-bounded by a given value. We develop different norms to measure the upgrade cost. Under weighted Hamming distance, we show the problems (SDIPT-UE/N) and (MCSDIPT-UE/N) are NP-hard by showing the equivalence of the two problems and the 0–1 knapsack problem. Under weighted l_1 norm, we solve the problems (SDIPT-UE) and (MCSDIPT-UE) in O(n) time by transforimg them into continuous knapsack problems. We propose two linear time greedy algorithms to solve the problem (SDIPT-UE) under unit Hamming distance and the problem (SDIPT-UN) with unit cost, respectively. Furthermore, for the the minimum cost problem (MCSDIPT-UE) under unit Hamming distance and the problem (MCSDIPT-UN) with unit cost, we provide two O(nlog n) time algorithms by the binary search methods. Finally, we perform some numerical experiments to compare the results obtained by these algorithms.

Models and algorithms for decomposition problems

This thesis deals with the decomposition both as a solution method and as a problem itself. A decomposition approach can be very effective for mathematical problems presenting a specific structure in which the associated matrix of coefficients is sparse and it is diagonalizable in blocks. But, this kind of structure may not be evident from the most natural formulation of the problem. Thus, its coefficient matrix may be preprocessed by solving a structure detection problem in order to understand if a decomposition method can successfully be applied. So, this thesis deals with the k-Vertex Cut problem, that is the problem of finding the minimum subset of nodes whose removal disconnects a graph into at least k components, and it models relevant applications in matrix decomposition for solving systems of equations by parallel computing. The capacitated k-Vertex Separator problem, instead, asks to find a subset of vertices of minimum cardinality the deletion of which disconnects a given graph in at most k shores and the size of each shore must not be larger than a given capacity value. Also this problem is of great importance for matrix decomposition algorithms. This thesis also addresses the Chance-Constrained Mathematical Program that represents a significant example in which decomposition techniques can be successfully applied. This is a class of stochastic optimization problems in which the feasible region depends on the realization of a random variable and the solution must optimize a given objective function while belonging to the feasible region with a probability that must be above a given value. In this thesis, a decomposition approach for this problem is introduced. The thesis also addresses the Fractional Knapsack Problem with Penalties, a variant of the knapsack problem in which items can be split at the expense of a penalty depending on the fractional quantity.

PPO-DFK: A Privacy-Preserving Optimization of Distributed Fractional Knapsack With Application in Secure Footballer Configurations

In the optimized footballer configurations, the team coach selects the players to participate in the game based on the training status of all players. As the number of excellent players increases, the costs and budget owned by the club boss need to be considered. Obviously, the players' costs are considered as private and sensitive and the training status are also sensitive. Therefore, the private information might be revealed in the process of data sharing and processing in this distributed manner. Considering the privacy-revealing issues in the above-mentioned scenario, this article proposes a privacy-preserving optimization for distributed fractional knapsack (PPO-DFK) problem, in which it achieves the secure footballer configurations, i.e., it is able to win the game but the sum of cost does not exceed the budget, without revealing either the expenditure/money owned by the club boss or the players' training status owned by the team coach. In the proposed PPO-DFK scheme, it employs a novel transformation approach (TA), a secure comparison protocol and a secure sorting protocol as the building blocks to ensure the privacy protection of distributed optimization procedure, then it uses the greedy algorithm to find an efficient solution. The security of proposed PPO-DFK scheme is strictly analyzed and its effectiveness is demonstrated by the experimental results on concrete examples.

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.

A novel three-phase target channel allocation scheme for multi-user Cognitive Radio Networks

Generation and allocation of Target Channel Sequence (TCS) in relation to spectrum handoff introduces several challenges in a multi-user Cognitive Radio Network. The primary challenge is to ensure fair and unique TCS allocation to multiple Secondary Users (SUs) in order to prevent channel access conflict during concurrent handoff operations. Another challenge is to perform dynamic TCS generation to guard against channel obsolescence effects under time-varying user activities. Inclusion of real-time SUs further mandates the use of time-bound TCS generation in order to prevent call drops due to time-outs during handoff. To address these issues of fairness, uniqueness, dynamic and time-bounded TCS generation, this paper proposes a novel machine-learning enabled three-phase TCS generation and allocation scheme for multiple real-time SUs involved in Voice over IP (VoIP) communication. While proactive and periodic phases compute unique TCS in O (n log n) time, the reactive phase performs on-demand TCS generation. Generation of TCS is formulated as a fractional knapsack problem and greedy technique is applied therein to optimally select channels based on their existing conditions. A novel HMM-BiLSTM (Hidden Markov Model-Bidirectional Long Short Term Memory) framework is further devised to safeguard VoIP interests through periodic channel eligibility prediction. Additionally, an analytical framework is developed which records negligible probability of channel access conflict among multiple SUs. Simulation results confirm fair and dynamic channel allocation with reduced handoff delays and call-drops and intelligent learning based decision-making operations with high accuracy in comparison to existing approaches. Finally, test-bed implementation validates practical applicability of the proposed methodology.

A Mean Field Game-Based Distributed Edge Caching in Fog Radio Access Networks

In this paper, the edge caching optimization problem in fog radio access networks (F-RANs) is investigated. Taking into account time-variant user requests and ultra-dense deployment of fog access points (F-APs), we propose a distributed edge caching scheme to jointly minimize the request service delay and fronthaul traffic load. Considering the interactive relationship among F-APs, we model the optimization problem as a stochastic differential game (SDG) which captures the dynamics of F-AP states. To address both the intractability problem of the SDG and the caching capacity constraint, we propose to solve the optimization problem in a distributive manner. Firstly, a mean field game (MFG) is converted from the original SDG by exploiting the ultra-dense property of F-RANs, and the states of all F-APs are characterized by a mean field distribution. Then, an iterative algorithm is developed that enables each F-AP to obtain the mean field equilibrium and caching control without extra information exchange with other F-APs. Secondly, a fractional knapsack problem is formulated based on the mean field equilibrium, and a greedy algorithm is developed that enables each F-AP to obtain the final caching policy subject to the caching capacity constraint. Simulation results show that the proposed scheme outperforms the baselines.

Characterization of the optimal solution of the convex separable continuous knapsack problem and related problems

A separable convex continuous knapsack problem with a single equality constraint and bounded variables is considered in this paper. Necessary and sufficient condition (characterization) for a feasible solution to be an optimal solution to this problem is stated, and characterization theorem in terms of a relaxed problem is formulated and proved. Versions of this problem with a single inequality constraint of the form “less than or equal to” and “greater than or equal to” are also considered, and sufficient conditions for solving these problems are stated and proved, based on the characterization theorem for the original problem. Examples of some convex separable objective functions for the considered problems are presented.

Manajemen Alokasi Bandwidth Layanan Internet Menggunakan Fractional Knapsack Problem

The technical problems faced in e-government implemented by the Ministry of Religion of Indonesia since 2015 are minimal bandwidth requirements to provide information services and behavior of users who access entertainment sites. When peak hours occur, the congested network often occurs which becomes a significant bottleneck. This study aims to implement bandwidth management using the fractional knapsack problem method by limiting access to entertainment services. The QoS parameters used in this management are throughput, delay, and jitter. The method was tested using a paired t-test using throughput, jitter, and delay test parameters by comparing test parameters before and after bandwidth management applied. The significance value produced is between 75-85%. The method used can control the amount of traffic for each service, but on the other, hand the delay and jitter are still high. It is necessary to add additional free space to each service that can be used when needed to reduce the delay and jitter.