Optimum design of composite shells subject to natural frequency constraints

A technique for the optimum (minimum weight) design of a composite shell subject to constraints on its natural frequencies is presented. The optimization problem is posed as a general mathematical programming problem in which one or more of the inequality constraints involves the shell natural frequencies, which must be evaluated numerically during the optimization. For this reason, a method for numerically evaluating the natural frequencies of composite shells is also presented. The method is based upon the finite element method of structural analysis and Rayleigh's principle. Because the element used is applicable to anisotropic shells of arbitrary shape, the method is very general. By using Rayleigh's principle, the necessity of assembling overall mass and stiffness matrices for the shell is eliminated. The optimization is performed by nondimensionalizing the mathematical programming problem and using the penalty function method of Fiacco and McCormick to transform the problem to a sequence of unconstrained minimizations having solutions which converge to the solution of the original (constrained) problem. The unconstrained minimizations are performed using the variable metric method of Fletcher and Powell. Derivatives of the nondimensional frequency constraints are evaluated numerically using difference equations. The frequency calculation method is demonstrated by calculating the fundamental frequency for the transverse vibration mode of a multilayered cylindrical shell with fixed overall geometry and variable composite geometry. Results indicate that the frequency increases with increasing fiber orientation angle, fiber volume fraction, or lamina thickness. The optimization technique is demonstrated by minimizing the weight of the shell discussed above subject to a constraint on its fundamental transverse frequency. The design variables are the fiber orientation angle, the fiber volume fraction, and the lamina thickness. Results are presented and explained in terms of the physical aspects of the problem.     

Formulations for the nonbifurcated hop-constrained multicommodity capacitated fixed-charge network design problem

This paper addresses the multicommodity capacitated fixed-charge network design problem with nonbifurcated flows and hop constraints. We present and compare mathematical programming formulations for this problem and we study different relaxations: Lagrangean relaxations, linear programming relaxations, and partial relaxations of the integrality constraints. In particular, we show that the Lagrangean bound obtained by relaxing the flow conservation equations is tighter than the linear programming relaxation bound. We present computational results on a large set of randomly generated instances.     

Linear 0–1 programming: A comparison of implicit enumeration algorithms

This paper compares the following implicit enumeration algorithms for solving a linear zero-one programming (LZOP) problem: Balas' additive algorithm, Hammer-Rudeanu's algorithm, Peterson's algorithm, Zionts' generalized additive algorithm, Geoffrion's improved implicit enumeration algorithm and Zionts' generalized additive algorithm with surrogate constraints. The computational efficiency of these algorithms is compared in terms of computer time and the number of iterations required to solve unstructured problems. Some guidelines for selecting an appropriate algorithm for a given problem size are given.     

Multi-item dynamic production-distribution planning in process industries with divergent finishing stages

This paper examines a multi-item dynamic production-distribution planning problem between a manufacturing location and a distribution center. Transportation costs between the manufacturing location and the distribution center offer economies of scale and can be represented by general piecewise linear functions. The production system at the manufacturing location is a serial process with a multiple parallel machines bottleneck stage and divergent finishing stages. A predetermined production sequence must be maintained on the bottleneck machines. A tight mixed-integer programming model of the production process is proposed, as well as three different formulations to represent general piecewise linear functions. These formulations are then used to develop three equivalent mathematical programming models of the manufacturer-distributor flow planning problem. Valid inequalities to strengthen these formulations are proposed and the strategy of adding extra 0–1 variables to improve the branching process is examined. Tests are performed to compare the computational efficiency of these models. Finally, it is shown that by adding valid inequalities and extra 0–1 variables, major computational improvements can be achieved.     

Optimizing player earnings in professional tennis

This paper devises strategies for a professional tennis player wishing to maximize earnings. First a computer program is developed, which predicts the probability of a player reaching successive rounds in a particular tournament, and also his expected earnings. This output provides the input variables to an integer programming problem. The optimal tournament set is selected by maximizing the player's expected earnings, subject to, constraints on the maximum number of tournaments to be played, the conflicting tournaments in a given week, and the travel constraints on concurrent tours composed of various tournaments.     

Models for the two‐dimensional rectangular single large placement problem with guillotine cuts and constrained pattern

In this paper, we address the constrained two‐dimensional rectangular guillotine single large placement problem (2D_R_CG_SLOPP). This problem involves cutting a rectangular object to produce smaller rectangular items from orthogonal guillotine cuts. In addition, there is an upper limit on the number of copies that can be produced of each item type. To model this problem, we propose a new pseudopolynomial integer nonlinear programming (INLP) formulation and obtain an equivalent integer linear programming (ILP) formulation from it. Additionally, we developed a procedure to reduce the numbers of variables and constraints of the integer linear programming (ILP) formulation, without loss of optimality. From the ILP formulation, we derive two new pseudopolynomial models for particular cases of the 2D_R_CG_SLOPP, which consider only two‐staged or one‐group patterns. Finally, as a specific solution method for the 2D_R_CG_SLOPP, we apply Benders decomposition to the proposed ILP formulation and develop a branch‐and‐Benders‐cut algorithm. All proposed approaches are evaluated through computational experiments using benchmark instances and compared with other formulations available in the literature. The results show that the new formulations are appropriate in scenarios characterized by few item types that are large with respect to the object's dimensions.     

Reactive power compensation by a linear programming technique

This article presents a method of optimizing the reactive compensation used in power systems to establish acceptable voltage profiles during period of abnormal loads and during foreseeable contingencies. The system equations, which are nonlinear, are first approximated to a linear form, and then linear programming technique is applied to obtain the quasi-optimal solution. An iterative procedure is then used to obtain results of acceptable accuracy. The main features of the proposed method are that both inductive and capacitative compensation is optimized and that the busbars where compensation is applied, can be selected to suit the user's operating constraints.     

On the exact solution of the no-wait flow shop problem with due date constraints

This paper deals with the no-wait flow shop scheduling problem with due date constraints. In the no-wait flow shop problem, waiting time is not allowed between successive operations of jobs. Moreover, the jobs should be completed before their respective due dates; due date constraints are dealt with as hard constraints. The considered performance criterion is makespan. The problem is strongly NP-hard. This paper develops a number of distinct mathematical models for the problem based on different decision variables. Namely, a mixed integer programming model, two quadratic mixed integer programming models, and two constraint programming models are developed. Moreover, a novel graph representation is developed for the problem. This new modeling technique facilitates the investigation of some of the important characteristics of the problem; this results in a number of propositions to rule out a large number of infeasible solutions from the set of all possible permutations. Afterward, the new graph representation and the resulting propositions are incorporated into a new exact algorithm to solve the problem to optimality. To investigate the performance of the mathematical models and to compare them with the developed exact algorithm, a number of test problems are solved and the results are reported. Computational results demonstrate that the developed algorithm is significantly faster than the mathematical models.     

A linear programming approach to shakedown analysis of structures

Two and three dimensional structures are dealt with, subjected to variable repeated loads, in order to establish a numerical tool for determining the load domain multiplier that gives rise to shakedown. The structure is made discrete by finite elements and the yield domain is linearized. By applying Bleich and Melan's theorem, two primal static formulations are found in linear programming, from which the relevant dual kinematic versions are obtained via duality properties.Numerical results are given at the end of the paper, together with some considerations about the numerical efficiency of the proposed formulations.     

Multi-media instructional approach to teaching linear programming

This paper describes different strategies employed in converting a lecture-oriented mathematical programming course to a Personalized Self-Paced Instructional (PSI) format. This is an elective course for students in science, engineering and management. A multi media instructional approach is used in the PSI system which combines traditional lectures, self-paced and individualized learning assisted by interactive computer programs and video taped instructional materials. This unique PSI system for mathematical programming provides maximum learning opportunity and flexibility to students. The author's experiences with the PSI system and the students' evaluation of the self-paced system are also discussed.     

Numerical modelling of pulsar magnetospheres

Numerical models used in the study of the pulsar magnetosphere are described: a vacuum model based only on Maxwell's equations and a more realistic model employing both Maxwell's and relativistic two-fluid equations. The general approach to solving the chosen sets of partial differential equations is outlined and the possible boundary conditions are examined. Numerical methods suitable for solving Maxwell's equations are discussed and a method is developed for solving the combined fluid plus Maxwell model. Results are presented and discussed and the possible improvements in the approach are indicated.     

Weighted residual methods in optimal control

Weighted residual methods (WRM) afford a viable approach to the numerical solution of differential equations. Application of WRM results in the transformation of differential equations into systems of algebraic equations in the modal coefficients. This suggests that WRM can be used as a tool for reducing optimal control problems to mathematical programming problems. Thereby, the optimal control problem is replaced by the minimization of a cost function of static coefficients subject to algebraic constraints. The motivation for this approach lies in the profusion of sophisticated computational algorithms and digital computer codes for the solution of mathematical programming problems. In this note the solution of optimal control problems as mathematical programming problems via WRM is illustrated. The example presented indicates that reasonable accuracy is obtained for modest computational effort. While the simplest types of modes-polynomials and piecewise constants-are employed in this note, the ideas delineated can be applied in conjunction with cubic splines for the generation of computational algorithms of enhanced efficiency.     

A NEW EXPERIMENTAL STUDY OF GENETIC ALGORITHM AND SIMULATED ANNEALING WITH BOUNDED VARIABLES

Genetic algorithms (GA) can work in very large and complex spaces, which gives them the ability to solve many complex real-world problems. The bounded variables linear programming is formulated as genetic algorithms and simulated annealing (SA). This article demonstrates that genetic algorithms and simulated annealing are much easier to implement for solving network problems compared with constructing mathematical programming formulations, because it is a very simple matter to implement a new cost function and solution constraints when using a GA and SA. Finally, the presented results show that the genetic algorithm and simulated annealing provide a good scheduling methodology to bounded variables programming.     

Application of mathematical programming in the design of optimal control systems†

The problem of applying the various computational methods of mathematical programming in the design of an optimal control system is discussed. A general case of non-linear, non-autonomous, state equations, subject to inequality constraints on both state and control variables, is considered. Both continuous and discrete time systems are investigated. In case of discrete time systems, the sampling intervals are assumed generally unequal and aperiodic, with inequality constraints imposed upon them.

Systems like these impose considerable computational difficulties when treated by the maximum principle or dynamie programming. Using mathematical programming, one may simplify a wide class of those computational problems.

Several examples of applying mathematical programming to particular control problems are presented.     

Direct and indirect boundary element methods for solving the heat conduction problem

The boundary element method is used to solve the stationary heat conduction problem as a Dirichlet, a Neumann or as a mixed boundary value problem. Using singularities which are interpreted physically, a number of Fredholm integral equations of the first or second kind is derived by the indirect method. With the aid of Green's third identity and Kupradze's functional equation further direct integral equations are obtained for the given problem. Finally a numerical method is described for solving the integral equations using Hermitian polynomials for the boundary elements and constant, linear, quadratic or cubic polynomials for the unknown functions.     

The use of optimization techniques in the analysis of cracked members by the finite element displacement and stress methods

Mathematical programming is applied to the two-dimensional stationary crack problem of a body composed of nonlinear elastic incompressible material. Fully admissible displacement as well as stress formulations are used to discretize the problem. Crack tip singularity is introduced in the displacement formulation by enriched elements for plane stress and, in certain cases, by superposition for plane strain. Pointwise incompressibility is obtained through constrained displacement functions. For three crack geometries Rice's J integral is evaluated by the energy difference method for different values of the hardening index. The numerical results, which are also applicable to secondary creep problems, appear to suggest a bounding character.     

A new mathematical programming approach to multi-group classification problems

In this paper we introduce a goal programming formulation for the multi-group classification problem. Although a great number of mathematical programming models for two-group classification problems have been proposed in the literature, there are few mathematical programming models for multi-group classification problems. Newly proposed multi-group mathematical programming model is compared with other conventional multi-group methods by using different real data sets taken from the literature and simulation data. A comparative analysis on the real data sets and simulation data shows that our goal programming formulation may suggest efficient alternative to traditional statistical methods and mathematical programming formulations for the multi-group classification problem.     

Some aspects of truss topology optimization

The present paper studies some aspects of formulations of truss topology optimization problems. The ground structure approach-based formulations of three types of truss topology optimization problems, namely the problems of minimum weight design for a given compliance, of minimum weight design with stress constraints and of minimum weight design with stress constraints and local buckling constraints are examined. The common difficulties with the formulations of the three problems are discussed. Since the continuity of the constraint or/and objective function is an important factor for the determination of the mathematical structure of optimization problems, the issue of the continuity of stress, displacement and compliance functions in terms of the cross-sectional areas at zero area is studied. It is shown that the bar stress function has discontinuity at zero crosssectional area, and the structural displacement and compliance are continuous functions of the cross-sectional area. Based on the discontinuity of the stress function we point out the features of the feasible domain and global optimum for optimization problems with stress and/or local buckling constraints, and conclude that they are mathematical programming with discontinuous constraint functions and that they are essentially discrete optimization problems. The difference between topology optimization with global constraints such as structural compliance and that with local constraints on stress or/and local buckling is notable and has important consequences for the solution approach.