Linear Programming Algorithm Research Articles

Solution of mathematical programming problems using tropical optimization methods

This paper deals with a class of mathematical programming problems that includes linear and nonlinear programming problems in a particular form. First, a linear programming problem is considered, and the possibility of deriving its direct complete solution in terms of traditional mathematics without using known iterative computational procedures and algorithms of linear programming, such as the simplex method, is studied. Direct solutions to the problem in the case of minimal dimension with a reduced set of constraints are proposed. It is shown that the derivation of such solutions, as dimension increases, becomes a very complicated problem with increasing dimension and, therefore, is hardly feasible. Some examples of other linear and nonlinear programming problems, which can be obtained from the above-considered problem by means of isomorphic transformations, are presented. The main definitions and preliminary results of tropical mathematics, which are required for the subsequent description and application of tropical optimization methods, are then outlined. A tropical optimization problem is formulated, and direct complete solutions of this problem and of its special cases are given. The above-formulated linear and nonlinear programming problems are reduced to a tropical optimization problem to provide their direct complete solution in terms of tropical mathematics. The solution of the linear programming problem with a reduced set of constraints is written in terms of traditional mathematics.

On learning and branching: a survey

This paper surveys learning techniques to deal with the two most crucial decisions in the branch-and-bound algorithm for Mixed-Integer Linear Programming, namely variable and node selections. Because of the lack of deep mathematical understanding on those decisions, the classical and vast literature in the field is inherently based on computational studies and heuristic, often problem-specific, strategies. We will both interpret some of those early contributions in the light of modern (machine) learning techniques, and give the details of the recent algorithms that instead explicitly incorporate machine learning paradigms.

Linear and dynamic programming algorithms for real-time task scheduling with task duplication

A real-time task scheduling system model was analyzed under a heterogeneous multiprocessor platform with task duplication. This analysis focused on the designs and performances of linear and dynamic programming algorithms for real-time task scheduling under a heterogeneous platform with task duplication. Moreover, experimental analyses were performed to evaluate the performances of different algorithms under different conditions. The advantages of the two proposed algorithms were compared under the same situations to discover which one achieves a higher task scheduling efficiency for a heterogeneous real-time system.

Model and algorithm for bilevel linear programming with fuzzy decision variables and multiple followers

Model and algorithm for bilevel linear programming with fuzzy decision variables and multiple followers

Model predictive control for inventory management in biomass manufacturing supply chains

This paper presents a centralised model predictive control strategy applied to biomass inventory control in sugarcane industries. Sugarcane industries are important renewable energy producers and an adequate inventory control of their feed material (biomass) can improve energy production. Simple linear discrete-time models with dead-time are developed to predict the controlled variable behaviour. Two layers are used in the controller, in the upper one performance is optimised by an linear programming (LP) algorithm and a multivariable generalised predictive controller (GPC) or multivariable generalised predictive controller with dead-time compensation (DTC-GPC) is used in the lower level. Simulation results in general show that the proposed controllers globally optimise the system behaviour and find an optimal ordering amount for keeping stock levels. In cases of plant/model mismatch DTC-GPC can have a significant and positive impact on the control of stock levels adding one more parameter for achieving minimised oscillatory performances (bullwhip effect).

An arc-search $${\mathcal {O}}(nL)$$ O ( n L ) infeasible-interior-point algorithm for linear programming

In this paper, we propose an arc-search infeasible-interior-point algorithm. We show that this algorithm is polynomial and the polynomial bound is ${\cal O}(nL)$ which is at least as good as the best existing bound for infeasible-interior-point algorithms for linear programming.

Gravity Sliding Algorithm for Linear Programming

An algorithm named Gravity Sliding is presented in the paper, which emulates the gravity sliding motion in a feasible region D constrained by a group of hyper planes in \(R^{m}\). At each stage point P of the sliding path, we need to calculate the projection of gravity vector g on constraint planes: if a constraint plane blocks the way at P, then it may change the direction of sliding path. The core technique is the synthetical treatment for multiple blocking planes, which is a basic problem of structural adjustment in practice; while the whole path provides the solution of a linear programming. Existing LP algorithms have no intuitive vision to emulate gravity sliding, therefore, their paths are not able to avoid circling and roving, and they could not provide a best direction at each step for structural adjustment. The first author presented the algorithm Cone Cutting (Wang in Inf Technol Decis Mak 10(1):65–82, 2011), which provides an intuitive explanation for Simplex pivoting. And then the algorithm Gradient Falling (Wang in Ann Data Sci 1(1):41–71, 2014. doi:10.1007/s40745-014-0005-9) was presented, which emulates the gradient motion on the feasible region. This paper is an improvement of gradient falling algorithm: in place of the description focusing on the null subspace of norm vectors, we focus the description on the expanding subspace of the very vectors in this paper. It makes the projection calculation easier and faster. We guess that the sliding path realized by the algorithm is the optimal path and the number of stage points of the path is limited by a polynomial function of the dimension number and the number of constraint planes.

Weekly maintenance scheduling using exact and genetic methods

The weekly maintenance schedule specifies when maintenance activities should be performed on the equipment, taking into account the availability of workers and maintenance bays, and other operational constraints. The current approach to generating this schedule is labour intensive and requires coordination between the maintenance schedulers and operations staff to minimise its impact on the operation of the mine. This paper presents methods for automatically generating this schedule from the list of maintenance tasks to be performed, the availability roster of the maintenance staff, and time windows in which each piece of equipment is available for maintenance. Both Mixed-Integer Linear Programming (MILP) and genetic algorithms are evaluated, with the genetic algorithm shown to significantly outperform the MILP. Two fitness functions for the genetic algorithm are also examined, with a linear fitness function outperforming an inverse fitness function by up to 5% for the same calculation time. The genetic algorithm approach is computationally fast, allowing the schedule to be rapidly recalculated in response to unexpected delays and breakdowns.

Dynamic prediction of financial distress using Malmquist DEA

Creditors such as banks frequently use expert systems to support their decisions when issuing loans and credit assessment has been an important area of application of machine learning techniques for decades. In practice, banks are often required to provide the rationale behind their decisions in addition to being able to predict the performance of companies when assessing corporate applicants for loans. One solution is to use Data Envelopment Analysis (DEA) to evaluate multiple decision-making units (DMUs or companies) which are ranked according to the best practice in their industrial sector. A linear programming algorithm is employed to calculate corporate efficiency as a measure to distinguish healthy companies from those in financial distress. This paper extends the cross-sectional DEA models to time-varying Malmquist DEA, since dynamic predictive models allow one to incorporate changes over time. This decision-support system can adjust the efficiency frontier intelligently over time and make robust predictions. Results based on a sample of 742 Chinese listed companies observed over 10 years suggest that Malmquist DEA offers insights into the competitive position of a company in addition to accurate financial distress predictions based on the DEA efficiency measures.

LP-based velocity profile generation for robotic manipulators

ABSTRACTIn this paper we examine the minimum-time velocity profile generation problem which belongs to the second stage of the decoupled robot motion planning. The time-optimal profile generation problem can be translated to a convex optimal control task through a nonlinear change of variables. When the constraints of the problem have special structure, the time-optimal solution can be obtained by linear programming (LP). In this special case, the velocity of the robot along the path is maximised, instead of time minimising. The benefit of the LP solution is the lower computational time. Validation of the LP algorithm is also presented based on simulation results.

Objective variation simplex algorithm for continuous piecewise linear programming

This paper works on a modified simplex algorithm for the local optimization of Continuous PieceWise Linear (CPWL) programming with generalization of hinging hyperplane objective and linear constraints. CPWL programming is popular since it can be equivalently transformed into difference of convex functions programming or concave optimization. Inspired by the concavity of the concave CPWL functions, we propose an Objective Variation Simplex Algorithm (OVSA), which is able to find a local optimum in a reasonable time. Computational results are presented for further insights into the performance of the OVSA compared with two other algorithms on random test problems.

An Improved SoC Test Scheduling Method Based on Simulated Annealing Algorithm

In this paper, we propose an improved SoC test scheduling method based on simulated annealing algorithm (SA). It is our first to disorganize IP core assignment for each TAM to produce a new solution for SA, allocate TAM width for each TAM using greedy algorithm and calculate corresponding testing time. And accepting the core assignment according to the principle of simulated annealing algorithm and finally attain the optimum solution. Simultaneously, we run the test scheduling experiment with the international reference circuits provided by International Test Conference 2002(ITC’02) and the result shows that our algorithm is superior to the conventional integer linear programming algorithm (ILP), simulated annealing algorithm (SA) and genetic algorithm(GA). When TAM width reaches to 48,56 and 64, the testing time based on our algorithm is lesser than the classic methods and the optimization rates are 30.74%, 3.32%, 16.13% respectively. Moreover, the testing time based on our algorithm is very close to that of improved genetic algorithm (IGA), which is state-of-the-art at present.

Economic scheduling model of microgrid considering the lifetime of batteries

Because of the uncertainty of renewable energy generation and load, batteries as energy storage devices play an important role in ensuring the safety and reliability of microgrid. To take full advantage of the batteries, the battery lifetime characteristics are analysed, and a weighted Wh throughput method is proposed to estimate the battery lifetime. To improve the economy of microgrid, an economic scheduling model of microgrid in grid-connected mode is established with the consideration of battery lifetime. For fast and efficiently solving the model, a technique is developed to convert the optimisation problem into a mixed integer linear programming problem and a mixed integer linear programming algorithm is applied to solve it. The proposed method has been validated on a microgrid, which consists of a wind turbine, a photovoltaic system, a micro turbine, a fuel cell, a diesel engine and a battery energy storage system (BESS). The simulation results show that the BESS is managed rationally and the total operation cost of microgrid over scheduling period is decreased by applying the proposed scheduling method.

A piecewise linear programming algorithm for sparse signal reconstruction

In order to recover a signal from its compressive measurements, the compressed sensing theory seeks the sparsest signal that agrees with the measurements, which is actually an l norm minimization problem. In this paper, we equivalently transform the l norm minimization into a concave continuous piecewise linear programming, and propose an optimization algorithm based on a modified interior point method. Numerical experiments demonstrate that our algorithm improves the sufficient number of measurements, relaxes the restrictions of the sensing matrix to some extent, and performs robustly in the noisy scenarios.

A Mixed Logical Dynamical-Model Predictive Control (MLD-MPC) Energy Management Control Strategy for Plug-in Hybrid Electric Vehicles (PHEVs)

Plug-in hybrid electric vehicles (PHEVs) can be considered as a hybrid system (HS) which includes the continuous state variable, discrete event, and operation constraint. Thus, a model predictive control (MPC) strategy for PHEVs based on the mixed logical dynamical (MLD) model and short-term vehicle speed prediction is proposed in this paper. Firstly, the mathematical model of the controlled PHEV is set-up to evaluate the energy consumption using the linearized models of core power components. Then, based on the recognition of driving intention and the past vehicle speed data, a nonlinear auto-regressive (NAR) neural network structure is designed to predict the vehicle speed for known driving profiles of city buses and the predicted vehicle speed is used to calculate the total required torque. Next, a MLD model is established with appropriate constraints for six possible driving modes. By solving the objective function with the Mixed Integer Linear Programming (MILP) algorithm, the optimal motor torque and the corresponding driving mode sequence within the speed prediction horizon can be obtained. Finally, the proposed energy control strategy shows substantial improvement in fuel economy in the simulation results.

$${\lambda }$$ λ -LGP: an improved version of linear genetic programming evaluated in the Ant Trail problem

The Ant Trail problem has been widely investigated as a benchmark for automatic design of algorithms. One must design the program of a virtual ant to collect all pieces of food located in different places of a map, which may have obstacles, in a predefined limit of steps. This is a challenging problem, but several evolutionary computation (EC) researchers have reported methods with good results. In this paper, we propose an EC method called \({\lambda }\)-linear genetic programming (\({\lambda }\)-LGP), a variation of the well-known linear genetic programming (LGP) algorithm. Starting with an LGP based only on effective macro- and micro-mutations, the \({\lambda }\)-LGP proposed in this work consists in extending how the individuals are chosen for reproduction. In this model, a number (\({\lambda }\)) of mutations is applied to each individual, trying to explore its neighboring fitness regions; such individual might be replaced by one of its children according to different criteria. Several configurations were tested over three different trails: the Santa Fe, the Los Altos Hill, and the John Muir. Results show a very significant improvement over LGP by using this proposed variation. Also, \({\lambda }\)-LGP outperformed not only LGP, but also other state-of-the-art methods from the literature.

The Watermelon Algorithm for The Bilevel Integer Linear Programming Problem

This paper presents an exact algorithm for the bilevel integer linear programming (BILP) problem. The proposed algorithm, which we call the watermelon algorithm, uses a multiway disjunction cut to remove bilevel infeasible solutions from the search space, which was motivated by how watermelon seeds can be carved out by a scoop. Serving as the scoop, a polyhedron is designed to enclose as many bilevel infeasible solutions as possible, and then the complement of this polyhedron is applied to the search space as a multiway disjunction cut in a branch-and-bound framework. We have proved that the watermelon algorithm is able to solve all BILP instances finitely and correctly, providing either a global optimal solution or a certificate of infeasibility or unboundedness. Computational experiment results on two sets of small- to medium-sized instances suggest that the watermelon algorithm could be significantly more efficient than previous branch-and-bound based BILP algorithms.

Geographical Infornational System for Assessment Spatial Distribution of Air Pollution (Case Study of Khmelnitsky City)

Purpose. The goal of research is spatial analysis of air quality in town Khmelnitsky and to reveal districts with the lowest air pollution level. Methods. Algorithms of purpose functions, linear programming and geoinformational modelling of spatial data. Results. The assessment of spatial distribution of air pollution in Khmelnitsky city has been implemented. The data base and appropriate geographical system have been developed. The geographical informational system includes all data concerning air pollution in Khmelnitsky city. There have been designed digital maps of air pollution the area in question.Conclusion. There have been revealed areas with the highest and lowest levels of air pollution.

Piezoelectric actuator design considering spillover effects

In this work, the topology optimization method using the solid isotropic material with penalization (SIMP) approach is employed to find the optimum design of piezoelectric actuators taking into account the excitation of the residual vibration modes. The structure governing equations are written into the state-space representation and the controllability Gramian eigenvalues are used to measure how the residual modes are susceptible to be excited by the control system. The proposed optimization formulation aims to find the distribution of piezoelectric material which maximizes the controllability for a given vibration mode while the undesirable effects of the feedback control on the residual modes are limited by including a spillover constraint term written as a p-norm of the residual controllability Gramian eigenvalues. The optimization of the shape and placement of the embedded piezoelectric actuators are carried out using a Sequential Linear Programming (SLP) algorithm. Numerical examples are presented for the control of the first bending vibration mode of a cantilever beam varying the set of residual modes considered in the spillover norm.

Hybrid Optimization Proposal for the Design of Collective On-rotation Operating Irrigation Networks

Hybrid models have been used in many engineering applications in order to find better solutions and to reduce project costs. In this paper a methodology for the optimal design of collective working shift irrigation systems is proposed. The proposal is based on a hybrid model of optimization, which includes Linear Programming (LP) and Genetic Algorithms (GA). The method is applied to an irrigation network to check its effectiveness to minimize the total investment costs for pipelines. The results are compared to another hybrid model for optimization, which is based on Nonlinear Programming (NLP) and GA. The advantage of the developed method is a more cost effective design using discrete pipeline diameters.