Sequential Linear Programming Method Research Articles

Application of Sequential Linear Goal Programming Method and Modified Simplex Method for Production Planning Optimization

Production planning is an important thing in company management. Every company aims to seek maximum profit in its business activities, so it must be able to utilize its resources as optimally as possible. This study was conducted to determine whether or not any production planning that has been implemented at Istana Roti Hilyah Bakery is optimal. This research aimed to maximize revenue and profit, minimize production costs and raw material usage and optimize production quantities using the sequential linear goal programming method and the simplex modification method with the assistance of LINGO software. Based on the study's results, the revenue increased by IDR 60,000, and the profit increased by IDR 66,679. The minimization value of production costs decreased by IDR 6,679, with the production amount of White Bread as many as 60 pcs, Almond Crumble Bread as many as 45 pcs, Chocolate Crumble Bread as many as 50 pcs, Wheat Bran Bread as many as 55 pcs, Sweet Comb Bread as many as 82 pcs and Jam Comb Bread as many as 84 pcs.

A Hilbertian projection method for constrained level set-based topology optimisation

We present an extension of the projection method proposed by Challis et al. (Int J Solids Struct 45(14–15):4130–4146, 2008) for constrained level set-based topology optimisation that harnesses the Hilbertian velocity extension-regularisation framework. Our Hilbertian projection method chooses a normal velocity for the level set function as a linear combination of (1) an orthogonal projection operator applied to the extended optimisation objective shape sensitivity and (2) a weighted sum of orthogonal basis functions for the extended constraint shape sensitivities. This combination aims for the best possible first-order improvement of the optimisation objective in addition to first-order improvement of the constraints. Our formulation utilising basis orthogonalisation naturally handles linearly dependent constraint shape sensitivities. Furthermore, use of the Hilbertian extension-regularisation framework ensures that the resulting normal velocity is extended away from the boundary and enriched with additional regularity. Our approach is generally applicable to any topology optimisation problem to be solved in the level set framework. We consider several benchmark constrained microstructure optimisation problems and demonstrate that our method is effective with little-to-no parameter tuning. We also find that our method performs well when compared to a Hilbertian sequential linear programming method.

Topology Optimization Design of Resonant Structures Based on Antiresonance Eigenfrequency Matching Informed by Harmonic Analysis

Abstract In this article, we present a design methodology for resonant structures exhibiting particular dynamic responses by combining an eigenfrequency matching approach and a harmonic analysis-informed eigenmode identification strategy. This systematic design methodology, based on topology optimization, introduces a novel computationally efficient approach for 3D dynamic problems requiring antiresonances at specific target frequencies subject to specific harmonic loads. The optimization’s objective function minimizes the error between target antiresonance frequencies and the actual structure’s antiresonance eigenfrequencies, while the harmonic analysis-informed identification strategy compares harmonic displacement responses against eigenvectors using a modal assurance criterion, therefore ensuring an accurate recognition and selection of appropriate antiresonance eigenmodes used during the optimization process. At the same time, this method effectively prevents well-known problems in topology optimization of eigenfrequencies such as localized eigenmodes in low-density regions, eigenmodes switching order, and repeated eigenfrequencies. Additionally, our proposed localized eigenmode identification approach completely removes the spurious eigenmodes from the optimization problem by analyzing the eigenvectors’ response in low-density regions compared to high-density regions. The topology optimization problem is formulated with a density-based parametrization and solved with a gradient-based sequential linear programming method, including material interpolation models and topological filters. Two case studies demonstrate that the proposed design methodology successfully generates antiresonances at the desired target frequency subject to different harmonic loads, design domain dimensions, mesh discretization, or material properties.

Global optimization of functional redundancy in a 6R robot for smoothing five-axis milling operations

Functional redundancy optimization is one of the most effective ways to improve robotic posture and performance. However, owing to the trade-off between smoothness of the robot's motion and other important aspects of performance, the optimization problem is too nonlinear to be solved in robotic milling processes. In this article, a derivation is shown that the robot's motion will be approximately smooth when the functional redundancies for each cutter location are smoothing variables. On this basis, a Legendre surrogate model is constructed to plan the overall optimal functional redundancy by remarkably reduced computation. Moreover, a trimmed sequential linear programming method is proposed to improve the solution further. From the outcomes of a 3D S-shaped grooves real milling experiment, the robot motion paths planned by the proposed methods achieve more stable tool-tip milling performance than that of comparable methods, while being several times or even 10 times faster in computational efficiency.

Unified Active and Reactive Power Coordinated Optimization for Unbalanced Distribution Networks in Radial and Looped Topology

Facing the high proportion of distributed generations incorporating in a single phase, the active distribution network has become more unbalanced with flexible topology. In this paper, a unified active and reactive power coordinated optimization (ARPCO) method, which is applicable in both radial and looped unbalanced distribution networks, is proposed. Aiming to reduce power losses and restrain undervoltage and overvoltage problems, the ARPCO model which regulates the active and reactive power output of distributed generations coordinately and optimally is constructed. A novel trust region sequential linear programming (SLP) method, which is effective in nonlinear and nonconvex model solving, is developed and employed in APRCO model solution. A multi-scenario case study based on the modified IEEE 123 node distribution system shows that the proposed method is able to reduce the system active power loss and solve undervoltage and overvoltage problems efficiently, at the same time maximizing the utilization of distributed generations.

Integration of Design for Additive Manufacturing Constraints With Multimaterial Topology Optimization of Lattice Structures for Optimized Thermal and Mechanical Properties

Abstract The design of multimaterial lattice structures with optimized elasticity tensor, coefficient of thermal expansion (CTE), and thermal conductivity is the main objective of the research presented in this article. In addition, the additive manufacturability of the lattice structure is addressed using a prismatic density filter to eliminate support structures, and an octant symmetry filter is used to design symmetric lattices. A density-based topology optimization model is formulated with a homogenization method and solved using a sequential linear programming method to obtain the desired unit cell geometry of the lattice structure. The optimized unit cell obtained has high mechanical stiffness, a low CTE, and low thermal conductivity. A finite element analysis is carried out on the optimized lattice structure and an equivalent cube of computed effective properties (with the same loading and boundary conditions) to validate the computed homogenized material properties. The results from the finite element analysis show that the methodology followed to generate the lattice structure is accurate. Such lattice structures with tailored material properties can be used in aerospace parts that are subjected to mechanical and thermal loads. The complex multimaterial geometry produced from the topology optimization routine presented here is intended explicitly for the manufacture of parts using the directed energy deposition process with multiple material deposition nozzles.

Multi-Material Topology Optimization Using Variable Density Lattice Structures for Additive Manufacturing

Multi-material lattice structures are used in a range of load-bearing applications for multiple conditions including mechanical and thermal loads. Additive manufacturing processes with multi-material capabilities are well suited to manufacture multi-material structures. In this paper, a multi-material topology optimization approach has been presented using variable-density lattice structures where the geometry of the lattice structure is pre-defined. The objective of the proposed topology optimization method is to design lightweight parts with minimized compliance and thermal energy or improve the heat transfer capability. To facilitate that, a novel interpolation scheme based on the stiffness matrices of the lattice structures has been proposed. This interpolation scheme, unlike the traditional Solid Isotropic Material Penalization (SIMP) interpolation, is observed to perform better in terms of approximating the structure’s load-bearing capacity, primarily due to its formulation on the lattice’s stiffness matrices. This cubic Hermite spline-based interpolation scheme makes it amenable for gradient-based optimization methods. A sequential linear programming method has been used to solve the weighted multi-objective optimization model. A Pareto-frontier study has also been carried out to fully characterize the trade-offs between the two objectives – compliance minimization and thermal energy minimization.

The tissue as self-assembly of notochord with sequential linear programming

Zebrafish serves as a model organism in morphogenesis studies in cell biology. One of the main difficulties of modeling a specific morphogenetic process come from the proper identification of its key features. To study the influence of geometric constraint in morphogenesis, here this article models the notochord as hard spheres assemble in an elliptical cylinder and optimize the packing density. This article implements linear approximation on the hard sphere constraints and find the optimal packing configuration by the sequential linear programming (SLP) method. During the process, this article modifies the SLP scheme into expanding the spheres in a fixed cylindrical confinement. This article finds the staircase and similar structures becomes predominant as increasing the ellipticity of the confinement. Our study deepens the understanding of the notochord morphogenesis, and also provides methodological guidance on related modeling studies involving geometric constraints.

Topology optimization of piezoelectric sensor and actuator layers for active vibration control

In this work, the layout of piezoelectric transducers is designed using the Topology Optimization Method (TOM). Starting with a finite element model and its modal reduction, a state-space model for a control scheme is built. With this model, two objective functions are formulated, i.e. the maximization of the trace of the controllability gramian matrix and the maximization of the trace of the observability gramian matrix. Each objective function simultaneously seeks an optimized distribution of piezoelectric material over an elastic structure. A SIMP material interpolation model, a spatial filter, and the Sequential Linear Programming (SLP) method with moving limits are used to solve the optimization problem. Moreover, a linear-quadratic-Gaussian controller (LQG) is implemented to verify the structural performance of a cantilever beam exposed to vibration. Finally, numerical results are shown for the transducers’ topologies and their control performance.

Solving nonconvex nonlinear programs with reverse convex constraints by sequential linear programming

AbstractThe sequential linear programming (SLP) method for solving nonlinear problems was introduced in the 1960s. Many papers that attempted to use SLP reported poor performance and convergence issues. We found that nonlinear programs with reverse convex constraints, which are the most difficult nonlinear programs with many local optima, are solved (heuristically) very well by SLP. We proved that for this type of problems, the solutions to the sequence of the linear programming problems converge to a local optimum. Since the final solution depends on the starting solution, we propose to apply SLP in a multistart approach starting from randomly generated solutions. This multistart SLP is very easy to implement. We recommend that the research community reconsiders the application of SLP for this type of problems.

Configuración óptima de transductores piezoeléctricos para control activo de vibraciones usando el método de optimización topológica

El método de optimización topológica (MOT) es usado para encontrar una distribución óptima de transductores piezoeléctricos (sensor y actuador) sobre una estructura elástica a partir de un modelo de elementos finitos convertido a variables en espacios de estado. Con dos funciones objetivo en paralelo, se busca maximizar la traza del gramiano de controlabilidad y la traza del gramiano de observabilidad. En la solución del problema de optimización se usa el modelo de interpolación de material SIMP y el método de Programación Lineal Secuencial (PLS); por tanto, se presenta el análisis de sensibilidad de ambas funciones objetivo. Finalmente, un controlador LQG es implementado para verificar el rendimiento estructural obtenido luego del proceso de optimización para un caso de estudio de una viga en Cantilever sometida a vibración, dispuesta de un actuador y un sensor piezoeléctrico. Los resultados de optimización son verificados con el análisis energético del esfuerzo de control, la sensibilidad del sensor, y el coeficiente de amortiguamiento del sistema planta-controlador.

Optimization of reinforced concrete polygonal sections under biaxial bending with axial force

Abstract Concrete is the world’s most utilized material for production of the structural elements employed in civil construction. Due to its low tensile strength and brittle nature it is reinforced with steel bars forming the reinforced concrete (RC structure). Linear elements of reinforced concrete are commonly employed in multi-story buildings, bridges, industrial sheds, among others. In this study an optimization algorithm is presented to define the amount of steel and its location within a concrete polygonal section subjected to biaxial bending with axial force, so that the amount of steel would be the minimum needed to resist the soliciting forces. Therefore, the project variables are: location, diameter and number of steel bars to be distributed within the concrete polygonal section. The sequential linear programming method is used to determine the optimized section. In this method, the non-linear problem of determining the resistance forces of the section in relation to the project variables is approximated by a sequence of linear problems, which would have its optimal point defined for each step using the Simplex method. Formulation validation is done through results of examples found in literature, and also by means of analytical solutions of simple problems, such as rectangular sections under axial force and moment in only one axis of symmetry. The results show the efficiency of the algorithm implemented in the optimized determination of the quantity and position of the bars of a given diameter in the polygonal section of reinforced concrete under biaxial bending with axial force.

An Efficient Topology Description Function Method Based on Modified Sigmoid Function

Optimal geometries extracted from traditional element-based topology optimization outcomes usually have zigzag boundaries, leading to being difficult to fabricate. In this study, a fairly accurate and efficient topology description function method (TDFM) for topology optimization of linear elastic structures is developed. By employing the modified sigmoid function, a simple yet efficient strategy is presented to tackle the computational difficulties because of the nonsmoothness of Heaviside function in topology optimization problem. The optimization problem is to minimize the structural compliance, with highest stiffness, while satisfying the volume constraint. The design problem is solved by a Sequential Linear Programming method. Convergent, crisp, and smooth final layouts are obtained, which can be fabricated without postprocessing, demonstrated by a series of numerical examples. Further, the proposed method has a rather higher accuracy and efficiency compared with traditional TDFM, when the classical topology optimization methods, such as bidirectional evolutionary structural optimization (BESO) and solid isotropic material with penalization (SIMP) method, are taken as benchmark.

Sequential linear programming and particle swarm optimization for the optimization of energy districts

This article deals with the optimization of energy resource management of industrial districts, with the aim of minimizing customer energy expenses. A model of the district is employed, whose optimization gives rise to a nonlinear constrained optimization problem. Here the focus is on its numerical solution. Two different methods are considered: a sequential linear programming method and a particle swarm optimization method. Efficient implementations of both approaches are devised and the results of the tests performed on several energetic districts are reported, including a real case study.

A Chance-Constrained Economic Dispatch Model in Wind-Thermal-Energy Storage System

As a type of renewable energy, wind energy is integrated into the power system with more and more penetration levels. It is challenging for the power system operators (PSOs) to cope with the uncertainty and variation of the wind power and its forecasts. A chance-constrained economic dispatch (ED) model for the wind-thermal-energy storage system (WTESS) is developed in this paper. An optimization model with the wind power and the energy storage system (ESS) is first established with the consideration of both the economic benefits of the system and less wind curtailments. The original wind power generation is processed by the ESS to obtain the final wind power output generation (FWPG). A Gaussian mixture model (GMM) distribution is adopted to characterize the probabilistic and cumulative distribution functions with an analytical expression. Then, a chance-constrained ED model integrated by the wind-energy storage system (W-ESS) is developed by considering both the overestimation costs and the underestimation costs of the system and solved by the sequential linear programming method. Numerical simulation results using the wind power data in four wind farms are performed on the developed ED model with the IEEE 30-bus system. It is verified that the developed ED model is effective to integrate the uncertain and variable wind power. The GMM distribution could accurately fit the actual distribution of the final wind power output, and the ESS could help effectively decrease the operation costs.

The Modified Sequential Linear Goal Programming Method for Solving Multiple Objectives Linear Programming Problems

Most of real world decision making problems have multiple objectives, which cannot be optimized simultaneously due to the conflicting nature of the objectives. Such problems can be solved by various methods to obtain the best-compromise solutions. Modified Sequential Linear Goal Programming (MSLGP) method can be used to solve Multiple Objective Linear programming Problems. In this paper, the use of existing single objective Linear Programming (LP) techniques is there, and the information required for MSLGP in each iteration are taken from the previous iteration. In this study, there is a great Revised Multi-phase Simplex Algorithm, which is used to solve MSLGP Accordingly within small number of computations as much as possible. This method is illustrated by some numerical examples, and provides ‘best compromise’ solution.

Optimal vaccination policy and cost analysis for epidemic control in resource-limited settings

Purpose – The purpose of this paper is to use analytical method and optimization tools to suggest time-optimal vaccination program for a basic SIR epidemic model with mass action contact rate when supply is limited. Design/methodology/approach – The Lagrange Multiplier Method and Pontryagin’s Maximum Principle are used to explore optimal control strategy and obtain analytical solution for the control system to minimize the total cost of disease with boundary constraint. The numerical simulation is done with Matlab using the sequential linear programming method to illustrate the impact of parameters. Findings – The result highlighted that the optimal control strategy is Bang-Bang control – to vaccinate with maximal effort until either all of the resources are used up or epidemic is over, and the optimal strategies and total cost of vaccination are usually dependent on whether there is any constraint of resource, however, the optimal strategy is independent on the relative cost of vaccination when the supply is limited. Practical implications – The research indicate a practical view that the enhancement of daily vaccination rate is critical to make effective initiatives to prevent epidemic from out breaking and reduce the costs of control. Originality/value – The analysis of the time-optimal application of outbreak control is of clear practical value and the introducing of resource constraint in epidemic control is of realistic sense, these are beneficial for epidemiologists and public health officials.

Material-Density-Based Topology Optimization With Magnetic Nonlinearity by Means of Stabilized Sequential Linear Programming: SLPSTAB

The nonlinear magnetic properties of an iron core should be considered when topology optimization (TO) is applied to a realistic problem. However, a scheme for introduction of the magnetic nonlinearity to material-density-based TO has not been rigorously established. In this paper, TO by using material density with magnetic nonlinearity is proposed. The numerical instability of the sequential linear programming method is a well-known topic. The stabilized sequential linear programming method, which realizes the stabilized convergence characteristics of an objective function, is proposed and investigated in the 3-D design problem of a magnetic circuit.

Five-axis flank milling of impellers: Optimal geometry of a conical tool considering stiffness and geometric constraints

Based on the Rayleigh–Ritz method, this article proposes an indicator to quantify the stiffness of a conical cutter for flank milling of impellers. Its validity is verified by the finite element analysis. A mathematical model is then developed to optimize the geometry of the conical cutter. The objective is to improve the stiffness of the cutter. Three kinds of geometric constraints are considered. First, the ball end of the cutter should be tangential to the hub surface. Second, the cutter should be interference-free with the adjacent blade. Finally, the machining error should satisfy the precision requirements. All these geometric constraints are characterized by the signed point-to-surface distance function. Based on the differential property of the distance function, a sequential linear programming method is applied to obtain the optimal geometry of the cutter along with the tool path. Simulation results confirm the effectiveness of the proposed model and algorithm.

Development and Field Test of Voltage VAR Optimization in the Korean Smart Distribution Management System

This paper is a summary of the development and demonstration of an optimization program, voltage VAR optimization (VVO), in the Korean Smart Distribution Management System (KSDMS). KSDMS was developed to address the lack of receptivity of distributed generators (DGs), standardization and compatibility, and manual failure recovery in the existing Korean automated distribution system. Focusing on the lack of receptivity of DGs, we developed a real-time system analysis and control program. The KSDMS VVO enhances manual system operation of the existing distribution system and provides a solution with all control equipment operated at a system level. The developed VVO is an optimal power flow (OPF) method that resolves violations, minimizes switching costs, and minimizes loss, and its function can vary depending on the operator’s command. The sequential mixed integer linear programming (SMILP) method was adopted to find the solution of the OPF. We tested the precision of the proposed VVO on selected simulated systems and its applicability to actual systems at two substations on the Jeju Island. Running the KSDMS VVO on a regular basis improved system stability, and it also raised no issues regarding its applicability to actual systems.