A mathematical programming model for production planning using CONWIP

Push-based MRP and pull-based Kanban systems are effective planning tools for a wide range of manufacturing production. Both of them, however, have certain limitations when they are implemented in different production environments. In recent years, CONWIP (CONstant Work-In-Process) has been proposed and studied to take advantage of MRP and Kanban systems for optimal work-inprocess (WIP) inventory control. In this paper, an integer nonlinear mathematical programming model was developed to determine an optimal production sequence and lot sizes in a CONWIP production line. The nonlinear programming model was linearized and solved directly for a number of illustrative example problems.     

Duality of regularizations and hullfunctions for mathematical programming problems

For the study of mathematical programming problems and solution methods the duality theory forms a powerful tool. There are also some concepts ofregularization andstabilization of a given problem for a better behavior in practical solution procedures. The aim of this paper is the investigation of duality aspects of such regularizations and the forming ofhullfunctions on the other hand. Applications for handling of so-calledill-posed problems (Eremin) using some parametrizations of the original problem will emphasize the importance for practical numerical methods, especially. This results will inspire some applications to solution methods for parametric and multicriteria optimization.     

LPL: A mathematical programming language

Summary This paper describes the new version of the modeling language, named LPL (Linear Programming Language). It may be used to build, modify and document mathematical models. The LPL language has been successfully applied to generate automatically MPS input files and reports of large LP models. The available LPL compiler translates LPL programs to the input code of any LP/MIP solver, calls the solver automatically, reads the solution back to its internal representation, and the integrated Report Generator produces the user defined reports of the model. Furthermore, an Input Generator can read the data from many formats.     

Multi-facility location problems in the presence of a probabilistic line barrier: a mixed integer quadratic programming model

We consider a multi-facility location problem in the presence of a line barrier with the starting point of the barrier uniformly distributed. The objective is to locate n new facilities among m existing facilities minimising the summation of the weighted expected rectilinear barrier distances of the locations of new facilities and new and existing facilities. The proposed problem is designed as a mixed-integer nonlinear programming model, conveniently transformed into a mixed-integer quadratic programming model. The computational results show that the LINGO 9.0 software package is effective in solving problems with small sizes. For large problems, we propose two meta-heuristic algorithms, namely the genetic algorithm and the imperialist competitive algorithm for optimisation. The numerical investigations illustrate the effectiveness of the proposed algorithms.     

Analysis and insights from a dynamical model of nuclear plant safety risk

In this paper, we expand upon previously reported results of a dynamical systems model for the impact of plant processes and programmatic performance on nuclear plant safety risk. We utilize both analytical techniques and numerical simulations typical of the analysis of nonlinear dynamical systems to obtain insights important for effective risk management. This includes use of bifurcation diagrams to show that period doubling bifurcations and regions of chaotic dynamics can occur. We also investigate the impact of risk mitigating functions (equipment reliability and loss prevention) on plant safety risk and demonstrate that these functions are capable of improving risk to levels that are better than those that are represented in a traditional risk assessment. Next, we analyze the system response to the presence of external noise and obtain some conclusions with respect to the allocation of resources to ensure that safety is maintained at optimal levels. In particular, we demonstrate that the model supports the importance of management and regulator attention to plants that have demonstrated poor performance by providing an external stimulus to obtain desired improvements. Equally important, the model suggests that excessive intervention, by either plant management or regulatory authorities, can have a deleterious impact on safety for plants that are operating with very effective programs and processes. Finally, we propose a modification to the model that accounts for the impact of plant risk culture on process performance and plant safety risk. We then use numerical simulations to demonstrate the important safety benefits of a strong risk culture.     

A closed formula for local sensitivity analysis in mathematical programming

This article introduces a method for local sensitivity analysis of practical interest. A theorem is given that provides a general and neat manner to obtain all sensitivities of a general nonlinear programming problem (around a local minimum) with respect to any parameter irrespective of it being a right-hand side, objective function or constraint constant. The method is based on the well-known duality property of mathematical programming, which states that the partial derivatives of the primal objective function with respect to the constraints' right-hand side parameters are the optimal values of the dual problem variables. For the parameters or data for which sensitivities are sought to appear on the right-hand side, they are converted into artificial variables and set to their actual values, thus obtaining the desired constraints. If the problem is degenerated and partial derivatives do not exist, the method also permits obtaining the right, left, and also directional derivatives, if they exist. In addition to its general applicability, the method is also computationally inexpensive because the necessary information becomes available without extra calculations. Moreover, analytical relations among sensitivities, locally valid, are straightforwardly obtained. It is also shown how the roles of the objective function and any of the active constraints (equality or inequality) can be exchanged leading to equivalent optimization problems. This permits obtaining the sensitivities of any constraint with respect to the parameters without the need of repeating the calculations. The method is illustrated by its application to two examples, one degenerated and the other one of a competitive market.     

Explicit model of dual programming and solving method for a class of separable convex programming problems

An objective function for a dual model of nonlinear programming problems is an implicit function with respect to Lagrangian multipliers. This study aims to address separable convex programming problems. An explicit expression with respect to Lagrangian multipliers is derived for the dual objective function. The exact solution of the dual model can be achieved because an explicit objective function is more exact than an approximated objective function. Then, a set of improved Lagrangian multipliers can be used to obtain the optimal solution of the original nonlinear programming model. A corresponding dual programming and explicit model (DP-EM) method is proposed and applied to the structural topology optimization of continuum structures. The solution efficiency of the DPEM is compared with the dual sequential quadratic programming (DSQP) method and method of moving asymptotes (MMA). The results show that the DP-EM method is more efficient than the DSQP and MMA.     

Lagrangean relaxation and constraint generation procedures for capacitated plant location problems with single sourcing

Summary Constraint generation procedures for identifying facet-induced inequalities violated by the optimal solution to the current LP relaxation have been widely used to solve integer programming problems. For capacitated plant location problems, Barcelo has recently tested computationally the performance of one such procedure. Hallefjord and Jörnsten have shown how these procedures can lead to better bounds when used with Lagrangean relaxations instead of the classical LP ones. We describe a Lagrangean relaxation heuristic algorithm for the capacitated location problem that in each iteration expands the dual space by adding to the dual Lagrangean function a new valid inequality for the problem generated from the current partial solution. Examples and computational results are included.

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Zusammenfassung Algorithmen zur Erzeugung von Nebenbedingungen zur Identifizierung von Facetteninduzierten Ungleichungen, die von der Optimallösung der aktuellen LP-Relaxation verletzt werden, werden häufig benutzt, um ganzzahlige Optimierungsprobleme zu lösen. Für kapazitierte Standortprobleme hat Barcelo kürzlich bereits rechnerisch die Güte eines solchen Algorithmusses getestet. Hallefjord und Jörnsten haben gezeigt, wie diese Algorithmen zu besseren Schranken führen können, wenn sie mit Lagrange-Relaxationen anstelle von klassischen LP's benutzt werden. Wir beschreiben einen heuristischen Lagrange-Relaxationsalgorithmus für das kapazitierte Standortproblem, der in jeder Iteration den dualen Raum erweitert, indem zur dualen Lagrange-Funktion eine neue, für das Ausgangsproblem gültige Ungleichungen addiert wird. Diese Ungleichung wird aus der aktuellen Teillösung erzeugt. Beispiele und numerische Ergebnisse sind angefügt.

     

Solving large-scale fixed cost integer linear programming models for grid-based location problems with heuristic techniques

Grid-based location problems (GBLPs) can be used to solve location problems in business, engineering, resource exploitation, and even in the field of medical sciences. To solve these decision problems, an integer linear programming (ILP) model is designed and developed to provide the optimal solution for GBLPs considering fixed cost criteria. Preliminary results show that the ILP model is efficient in solving small to moderate-sized problems. However, this ILP model becomes intractable in solving large-scale instances. Therefore, a decomposition heuristic is proposed to solve these large-scale GBLPs, which demonstrates significant reduction of solution runtimes. To benchmark the proposed heuristic, results are compared with the exact solution via ILP. The experimental results show that the proposed method significantly outperforms the exact method in runtime with minimal (and in most cases, no) loss of optimality.     

A study of mathematical programming methods for structural optimization. Part I: Theory

A comprehensive study of various mathematical programming methods for structural optimization is presented. In recent years, many modern optimization techniques and convergence results have been developed in the field of mathematical programming. The aim of this paper is twofold: (a) to discuss the applicability of modern optimization techniques to structural design problems, and (b) to present mathematical programming methods from a unified and design engineers' viewpoint. Theoretical aspects are considered here, while numerical results of test problems are discussed in a companion paper. Special features possessed by structural optimization problems, together with recent developments in mathematical programming (recursive quadratic programming methods, global convergence theory), have formed a basis for conducting the study. Some improvements of existing methods are noted and areas for future investigation are discussed.     

A mixed integer programming model for optimal ATC tower height and location: a case study for Singapore Changi Airport's third runway extension

ABSTRACT

With continued growth in air traffic, airports worldwide are expanding their runway infrastructure. This leads to the problem of determining an appropriate location and height for an Air Traffic Control (ATC) tower that can provide the right vantage point for coordinating runway and taxiway movements. The challenge involves finding the right location and optimal height that can satisfy the visibility and obstruction constraints for a complex airport-airside environment with multiple runways and civil infrastructure under different weather conditions. This article formulates the ATC tower location and height problem as a Mixed-Integer-Programming (MIP) model while considering the visibility and obstruction constraints. Singapore Changi Airport's proposed third runway extension is used as a case study to determine the set of location and height of ATC Tower using the proposed approach. A visual analytic test is conducted in an ATC tower simulator for different tower locations and heights under varying visibility conditions.     

Production outsourcing: A linear programming model for the Theory-Of-Constraints

This paper presents an analysis of the outsourcing problem. Pertinent variables are identified and the relationships between them are defined. We formulate the outsourcing problem as a Linear-Programming (LP) problem and identify an analytical solution. We proceed with an example examining three decision models: standard cost accounting, standard Theory-Of-Constraints (TOC) and our own solution. The model enables managers to determine which products to manufacture and which to outsource. The solution of the LP formulation enables managers to apply the model by computing an operational ratio, without having to solve a linear programming problem. The final model is simpler to apply and requires the computation of fewer variables than other prevalent models.     

A mathematical programming method for the inelastic analysis of reinforced concrete frames

A mathematical programming method is presented for solving problems of (i) determining the moments and rotations and (ii) determining the safety factor of reinforced concrete frames. The proposed method is essentially a reformulation of those developed by De Donato and Maier. However, the reformulation results in a new mathematical programming model for which an efficient algorithm is developed. The advantage of the proposed method is mainly computational; in fact it requires roughly one half computer time and storage space as compared with those De Donato and Maier employ to solve the same problem.     

多核集群任务分配问题的0-1整数规划求解模型

研究了典型多核集群任务分配中的节点内通讯特性。基于0-1整数非线性规划模型和线性松弛技术,给出了一种0-1整数线性规划任务分配问题求解优化模型。由于节点内的通讯量与通讯延迟较大,以最小化计算代价和节点间通讯代价为研究目标的传统求解模型具有严重的局限性,而该求解模型考虑了节点内通讯代价,并采用了线性规划松弛技术,其目标是最小化计算代价、节点间通讯代价和节点内通讯代价。计算结果验证了提出的模型的有效性。     

一个核反应堆的数学模型的整体解的渐近性质

文章用 Lp估计一个核反应堆的数学模型整体解的渐近性 ,并对主要结果作了相应的证明 .     

A mathematical model for pulmonary mechanics: The alveolar surface contribution

A rationally motivated, one degree of freedom, scalar, mathematical model is developed from a statement of conservation of mass and surface thermodynamics to represent the contribution of alveolar surface tension to the mechanics of respiration. In the course of development of the model, a relationship for dynamic surface tension as a function of area is obtained. Surface free energy per unit area vs area diagrams are presented. The model predicts spatial average values of the transalveolar-surface pressure as a function of time for the whole lung, not necessarily true at any specific alveoli at that time. The present value of pressure is a function not only of the present lung volume, but also the volume history. The transpulmonary pressure is formulated in terms of the pressure due to airway resistance, that of deformation of lung tissue, and the transalveolar surface pressure. New lung constants arise in a natural way from the model. The pulmonary mechanics predicted by the model is consistent with the concept that normal lung alveolar surface tension is below the thermostatic equilibrium value and is maintained in such a state by occasional deep inspirations.     

A model for the growth of mathematical specialties

A mathematical model for the growth of two coupled mathematical specialties, differential geometry and topology, is analyzed. The key variable is the number of theorems in use in each specialty. Obsolescences of theorems-in-use due to replacement by more general theorems introduces non-linear terms of the differential equations. The stability of stationary solutions is investigated. The phase portrait shows that the number of theorems in low-dimensional topology relative to those in differential geometry is increasing. The model is qualitatively consistent with the growth of publications in these two specialties, but does not give quantitative predictions, partly because we do not use an explicit solutions as a function of time and partly because only two specialties are used. The methods of analysis and some of the concepts can be extended to the development of more general and realistic models for the growth of specialties.Supported in part by grant IST 78-16629. The authors would like to express their thanks to Derek deSolla Price for very valuable comments that helped us to improve this paper.     

A mathematical model of saltation

Summary This paper presents a model of the uplift by wind of heavy (i.e. saltating and unaffected by turbulent diffusion) soil particles. Detailed trajectories are not calculated, although trajectory heights are required and this necessitates taking account of the drag on particles. Quantities required as input are (a) the mass-concentration at a height equal to the roughness length above the surface for each size class, (b) the roughness length of the undisturbed surface, and (c) the friction velocity which is assumed to be controlled by the overlying wind and unaffected by erosion. The model is combined with a previously bublished model for lighter particles and applied to hypothetical size distributions for a range of friction velocities. Mean-diameter profiles and horizontal fluxes are obtained and found to agree well with observations by various workers.