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.     

A mathematical programming model for reconfiguration of flexible manufacturing cells

In flexible manufacturing cells, changes in demand size, product mix, part variety, existing routings and set-up/operation times may require reconfiguration of the cells. In this study, an approach is developed to decide when and how such a reconfiguration should be carried out for existing cells. This study considers reconfiguration in terms of changing part routings, adding a new machine type in a cell, duplicating an existing machine type, removing an existing machine from a cell and transferring a machine to another cell. The study also shows the total number of tools of each type on each machine located in each cell after reconfiguration. To make the optimal reconfiguration decisions, a mathematical programming model to minimize the total reconfiguration cost is developed. The developed model considers the lower and upper bounds on machine utilizations and the time limits on machine cycle times to decide when to reconfigure the system.     

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.     

A new deterministic global optimization method for general twice-differentiable constrained nonlinear programming problems

A deterministic global optimization method that is applicable to general nonlinear programming problems composed of twice-differentiable objective and constraint functions is proposed. The method hybridizes the branch-and-bound algorithm and a convex cut function (CCF). For a given subregion, the difference of a convex underestimator that does not need an iterative local optimizer to determine the lower bound of the objective function is generated. If the obtained lower bound is located in an infeasible region, then the CCF is generated for constraints to cut this region. The cutting region generated by the CCF forms a hyperellipsoid and serves as the basis of a discarding rule for the selected subregion. However, the convergence rate decreases as the number of cutting regions increases. To accelerate the convergence rate, an inclusion relation between two hyperellipsoids should be applied in order to reduce the number of cutting regions. It is shown that the two-hyperellipsoid inclusion relation is determined by maximizing a quadratic function over a sphere, which is a special case of a trust region subproblem. The proposed method is applied to twelve nonlinear programming test problems and five engineering design problems. Numerical results show that the proposed method converges in a finite calculation time and produces accurate solutions.     

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.     

A new mathematical model for the Weber location problem with a probabilistic polyhedral barrier

With the wide application of location theory in a variety of industries, the presence of barriers ?merits the attention of managers and engineers.? In this paper, we assess the Weber location problem in the presence of a polyhedral barrier which probabilistically occurs on a given horizontal barrier route in the rectilinear space. A left triangular distribution function is used for the starting point of the barrier and therefore an expected rectilinear barrier distance function is formulated. In addition, a modification of the polyhedral barrier is presented which is equivalent to the original problem. Therefore, a mixed integer nonlinear programming model, which has a nonconvex solution space, is presented. Furthermore, by decomposing the feasible space into a finite number of convex solution spaces, an exact heuristic solution method is proposed. Then, a lower bound problem based on the forbidden region is applied. Some theorems and an example are reported.     

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 dynamical system perspective on mathematical programming

We explore how to build a vector field from the various functions involved in a given mathematical programme, and show that equilibria of the corresponding dynamical system are precisely the solutions of the underlying optimality conditions for the original optimization problem. The general situation in which explicit inequality constraints are present is especially interesting as the vector field has to be discontinuous, and so one is led to consider discontinuous dynamical systems and their equilibria.     

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.     

Solving 2D transient rolling contact problems using the BEM and mathematical programming techniques

This work presents a new approach to the transient rolling contact of two‐dimensional elastic bodies. A solution will be obtained by minimizing a general B‐differentiable function representing the equilibrium equations and the contact conditions at each time step. Inertial effects are not taken into account and the boundary element method is used to compute the elastic influence coefficients of the surface points involved in contact (equilibrium equations). The contact conditions are represented with the help of variational inequalities and projection functions. Finally, the minimization problem is solved using the Generalized Newton's Method with line search. The results are compared with some example problems and the influence of discretization and integration time step on the results is discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Fuzzy analysis for steady-state availability: a mathematical programming approach

Steady-state availability has been widely applied as a measure to evaluate the reliability characteristics of a repairable system. However, it is generally not realistic to make assumptions concerning failure time and repair time distributions. Thus, this article has developed a procedure to construct the membership function for fuzzy steady-state availability. Based on Zadeh’s extension principle, a pair of mathematical programs is formulated to find α-cuts of fuzzy steady-state availability. An explicit closed-form expression for the membership function is derived by taking the inverse function of the $\alpha$-cut. To illustrate the interpretation and practical value of fuzzy availability in real-world applications, several numerical examples are provided and discussed.     

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.     

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.

     

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.