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Two main goals are pursued in the present work. First, the global buckling of axially loaded sandwich beams with composite face sheets and aluminum honeycomb cores under humid environments is investigated. An analytical solution with high accuracy is developed based on piecewise low-order shear deformation theory (PLSDT) to estimate the critical buckling loads. A parametric study is carried out to examine the influence of geometric parameters on the buckling loads of simply supported and clamped sandwich beams under different wet conditions. The obtained results indicate that humidity plays a significant role in the buckling behavior of honeycomb sandwich beams. As the second purpose of the present work, the weight minimization of sandwich beams is studied. The optimization variables include the face sheet thickness, core thickness, honeycomb cell size and honeycomb cell wall thickness. Moreover, different failure modes, including global buckling, face sheet dimpling, face sheet wrinkling, core shear instability, and face sheet failure are considered as design constraints. Imperialist competitive algorithm (ICA), a powerful socio-politically motivated global search strategy, is applied to handle the constrained optimization problem. The optimization results show that the presence of humidity in the environment significantly changes the optimal design of sandwich beams and makes them heavier. Furthermore, it is observed that by increasing the length of the structure, the optimal design is more affected by humidity. Moreover, it is found that ICA is a very powerful optimization tool both in obtaining the best values of geometric parameters and in reducing the computational cost. [ABSTRACT FROM AUTHOR]

In recent years, increased sales of fuel cell electric vehicles (FCEVs) have required composite overwrapped pressure vessel (COPV) designs to be lightweight and allow safe high-pressure hydrogen storage. In this study, we propose the weight minimization of Type 2 COPVs for FCEVs considering mechanical safety. Steel liner thickness, ply thickness, ply orientation, and the number of plies were set as design variables, and weight minimization was performed. For the constraints of optimization, the Tsai–Wu failure index of the composite layer and von Mises stress of the steel liner are considered. The design of experiments (DoE) was conducted to generate kriging model and perform sensitivity analysis. The optimized design of Type 2 COPVs was determined by satisfying all constraints, with significant weight reduction and preserved mechanical safety of the structure. [ABSTRACT FROM AUTHOR]

In engineering practice, one can often encounter issues related to optimization, where the goal is to minimize material consumption and minimize stresses or deflections of the structure. In most cases, these issues are addressed with finite element analysis software and simple optimization algorithms. However, in the case of optimization of certain structures, it is not so straightforward. An example of such constructions are bubble deck ceilings, where, in order to reduce the dead weight, air cavities are used, which are regularly arranged over the entire surface of the ceiling. In the case of these slabs, the flexural stiffness is not constant in all its cross-sections, which means that the use of structural finite elements (plate or shell) for static calculations is not possible, and therefore, the optimization process becomes more difficult. This paper presents a minimization procedure of the weight of bubble deck slabs using numerical homogenization and sequential quadratic programming with constraints. Homogenization allows for determining the effective stiffnesses of the floor, which in the next step are sequentially corrected by changing the geometrical parameters of the floor and voids in order to achieve the assumed deflection. The presented procedure allows for minimizing the use of material in a quick and effective way by automatically determining the optimal parameters describing the geometry of the bubble deck floor cross-section. For the optimal solution, the concrete weight of the bubble deck slab was reduced by about 23% in reference to the initial design, and the serviceability limit state was met. [ABSTRACT FROM AUTHOR]

In recent years, increased sales of fuel cell electric vehicles (FCEVs) have required composite overwrapped pressure vessel (COPV) designs to be lightweight and allow safe high-pressure hydrogen storage. In this study, we propose the weight minimization of Type 2 COPVs for FCEVs considering mechanical safety. Steel liner thickness, ply thickness, ply orientation, and the number of plies were set as design variables, and weight minimization was performed. For the constraints of optimization, the Tsai–Wu failure index of the composite layer and von Mises stress of the steel liner are considered. The design of experiments (DoE) was conducted to generate kriging model and perform sensitivity analysis. The optimized design of Type 2 COPVs was determined by satisfying all constraints, with significant weight reduction and preserved mechanical safety of the structure.

The inverse motion concept is used to optimize thermo-hyperelastic structures using an exact description of the deformed geometry. This method prescribes the shape of the structure in the deformed state, and the optimization yields the shape of the undeformed configuration, i.e. the manufactured state. The kinematics of the thermoelastic model is defined through the multiplicative decomposition of the deformation gradient in combination with neo-Hookean hyperelasticity. To regularize the optimization problem and obtain distinct boundaries, the mathematical design field is thresholded using a smoothed Heaviside function and smeared using a partial differential equation. The sensitivity analyses of the objective function and constraints are both based on the adjoint method. The capabilities of the proposed approach are shown by numerical examples wherein the weight is minimized and the performance of multi-material compliant mechanisms is optimized. [ABSTRACT FROM AUTHOR]

The present work addresses the optimal design of sandwich panels made of flax fabric (FF)/vinyl ester (VE) composite face sheets and honeycomb VE core. The sandwich structures are first optimized in terms of flammability by obtaining the best combination of ammonium polyphosphate (APP), halloysite nanotube (HNT), and magnesium hydroxide (MH) as three flame retardants (FRs). Using the Taguchi method and horizontal burning test, it is shown that [6, 3, and 3%] and [1, 0.5, and 0%] are the optimal combinations of APP, HNT, and MH for the face sheets and core, respectively. Cone calorimeter test results indicate that the optimal FR combinations significantly decrease the mass lost rate (MLR), heat rate release (HRR), total smoke release (TSR), and maximum average release heat emission (MARHE). The FR sandwich structures are then geometrically optimized under compressive loads based on their weight. Different failure modes are considered as the design constraints of the optimization problem. Imperialist competitive algorithm (ICA), as a powerful meta-heuristic algorithm, is implemented to considerably reduce the computational cost of the optimization process. The results of this study show that proper combinations of FR additives can increase the flame retardancy while decreasing the weight of sandwich panels.

In engineering practice, one can often encounter issues related to optimization, where the goal is to minimize material consumption and minimize stresses or deflections of the structure. In most cases, these issues are addressed with finite element analysis software and simple optimization algorithms. However, in the case of optimization of certain structures, it is not so straightforward. An example of such constructions are bubble deck ceilings, where, in order to reduce the dead weight, air cavities are used, which are regularly arranged over the entire surface of the ceiling. In the case of these slabs, the flexural stiffness is not constant in all its cross-sections, which means that the use of structural finite elements (plate or shell) for static calculations is not possible, and therefore, the optimization process becomes more difficult. This paper presents a minimization procedure of the weight of bubble deck slabs using numerical homogenization and sequential quadratic programming with constraints. Homogenization allows for determining the effective stiffnesses of the floor, which in the next step are sequentially corrected by changing the geometrical parameters of the floor and voids in order to achieve the assumed deflection. The presented procedure allows for minimizing the use of material in a quick and effective way by automatically determining the optimal parameters describing the geometry of the bubble deck floor cross-section. For the optimal solution, the concrete weight of the bubble deck slab was reduced by about 23% in reference to the initial design, and the serviceability limit state was met.

The performance of majority engineering systems made of composite laminates can be improved by increasing strength to weight ratio. Variable thickness approach (VTA), in discrete form, used in this study is capable of finding minimum laminate thickness in one stage only, instead of two stage methodology defined by other researchers, with substantial accuracy for the given load conditions. This minimum required laminate thickness can be used by designers in multiple ways. Current study reveals that effectiveness of VTA in this regard depends on ply thickness increment value and number of plies. Maximum Stress theory, Tsai Wu theory and Tsai Hill theory are used as constraints, while ply angles, ply thicknesses and number of plies in discrete form are used as design variables in current simulation studies. Optimization is carried out using direct value coded genetic algorithm. The effect of design variables such as ply angles, ply thicknesses and number of plies in discrete form on optimum solution is investigated considering Uniform Thickness Approach (UTA) and Variable Thickness Approach (VTA) for various load cases.

This work presents a methodology for the design of a hydrogen fuel cell-based hybrid electric power plant for hybrid electric vehicles (HEV), where a battery bank and ultracapacitors are also considered as components of the hybrid power plant. The methodology considers the design features of an electric vehicle and evaluates its energy and power requirements as to fulfil a driving cycle. The work starts by weight minimizing a fuel cell taking into consideration its physical and electrochemical characteristics. Batteries and ultracapacitors are then sized according to their dynamic response features and considering specifications from commercial candidate cells, to propose an electric configuration and specify the baseline for a hybrid power plant. In order to illustrate the methodology, a crossover utility electric vehicle and a WLTC class I drive cycle are used. This work shows that by reducing the power plant size, power and energy requirements can also be minimized and the overall performance can be increased promoting fuel and costs savings. For comparison and to show the impact of weight minimization on the energy on board and cost, this work presents the energy and power required by different power plant configurations. Results showed that including ultracapacitors to the power plant offers more benefits, such as less stress on batteries, at a marginal initial cost compared to a case without ultracapacitors, where batteries should attend transients with a limited capability for energy recovery from regenerative breaking. The methodology is easily implemented and does not large computational resources providing with a power plant baseline for further design stages, such as particular energy management approaches depending on particular priorities for the developer, such as range, productivity and performance, economy and others. • A fuel cell design as the main energy source of a hybrid electric vehicle. • Sizing method of energy sources independent of associated energy management strategy. • Sizing aiming at minimizing hybrid electric power plant weight. • Sources size reference values for any energy management strategy. [ABSTRACT FROM AUTHOR]

The aim of this study is to use stochastic optimization algorithm MSAA (Modified Simulated Annealing Algorithm) for trusses plane optimization (weight minimization) with discrete variables. MSAA is based on the cooling process of metal used in the Simulated Annealing (SA) classic, but it has three fundamental characteristics (preliminary exploration, search step and acceptance probability) that differentiate this. To evaluate and validate the MSAA performance were studied three problems plane trusses weight minimization with discrete variables reported in the literature and the results are compared with those obtained by other authors using different optimization algorithms. It is concluded that the MSAA algorithm presented in this study can be effectively used in the weight minimization of truss structures.

Here, analytical studies have been carried out to determine the optimal values of effective parameters on the stress concentration factor around a cutout using genetic algorithm. Optimum designs of single lamina as well as symmetric laminates with 4, 8 and 12 layers of graphite/epoxy and glass/epoxy plates containing a circular cutout with various sizes are presented. The work focuses on extending the analytical solution given by Greszczuk to determine the stress distribution in multilayered composite plates subjected to arbitrary in-plane loadings. This is achieved by introducing an arbitrary oriented uniaxial, biaxial and shear loading conditions into Greszczuk solution. In order to mimic as much as possible the real structural behavior, the finite-width correction factor given by Tan is used. Effective parameters on stress distribution around the circular cutout in composite plates considered as design variables include: load angle, fiber orientation, cutout size and stacking sequence of the laminate. The objective function in this study is the minimization of maximum stress concentration around the cutout which is calculated by the present analytical solution. The first ply failure load predicted using Tsai–Wu criterion is maximized for both single lamina and symmetric laminates. Also, the weight of the plates is minimized by increasing the hole size to width ratio. The results obtained by the present analytical solution compare favorably with those obtained using complex variable method. For laminated plate subjected to shear loading, the stress concentration factor decreased drastically by 48.79% compared to a single lamina. The failure load is also increased in most of the loading cases. The results also showed that the genetic algorithm code converges rapidly in most of the cases. The accuracy, quickness, low computational cost and the simplicity of the present solution encourage the designers to use it in practical applications. [ABSTRACT FROM AUTHOR]

The performance of majority engineering systems made of composite laminates can be improved by increasing strength to weight ratio. Variable thickness approach (VTA), in discrete form, used in this study is capable of finding minimum laminate thickness in one stage only, instead of two stage methodology defined by other researchers, with substantial accuracy for the given load conditions. This minimum required laminate thickness can be used by designers in multiple ways. Current study reveals that effectiveness of VTA in this regard depends on ply thickness increment value and number of plies. Maximum Stress theory, Tsai Wu theory and Tsai Hill theory are used as constraints, while ply angles, ply thicknesses and number of plies in discrete form are used as design variables in current simulation studies. Optimization is carried out using direct value coded genetic algorithm. The effect of design variables such as ply angles, ply thicknesses and number of plies in discrete form on optimum solution is investigated considering Uniform Thickness Approach (UTA) and Variable Thickness Approach (VTA) for various load cases. [ABSTRACT FROM AUTHOR]

Topology optimization is one of the most effective tools for conducting lightweight design and has been implemented across multiple industries to enhance product development. The typical topology optimization problem statement is to minimize system compliance while constraining the design space to an assumed volume fraction. The traditional single-material compliance problem has been extended to include multiple materials, which allows increased design freedom for potentially better solutions. However, compliance minimization has the limitations for practical lightweight design because compliance lacks useful physical meanings and has never been a design criterion in industry. Additionally, the traditional compliance minimization problem statement requires volume fraction constraints to be selected a priori; however, designers do not know the optimized balance among materials. In this paper, a more practical method of multi-material topology optimization is presented to overcome the limitations. This method seeks the optimized balance among materials by minimizing the total weight while satisfying performance constraints. This paper also compares the weight minimization approach to compliance minimization. Several numerical examples prove the success of weight minimization and demonstrate its benefit over compliance minimization. [ABSTRACT FROM AUTHOR]

The article concerns optimization of network arch bridges. This is challenging optimization problem involving even for conventional scheme of network arch bridge the identification of some topological parameters as well as shape configurations and all sizing parameters of structural members, seeking the minimum weight. Optimal bridge scheme is sought tuning a large set of design parameters of diverse character: the type of hanger arrangement, the number of hangers, their inclination angles and placement distances, the arch shape and rise, etc. Mathematically, the optimization of the bridge scheme is a mixed-integer constrained global optimization problem solved employing stochastic evolutionary algorithm. Plane heavy/ moderate/and light-deck bridges of 18, 30, 42 and 54 m spans were optimized using proposed optimization technique. The decisive design parameters and their promising ranges were revealed. Also, the influence of some simplifications is shown: changing the arch shape from elliptical to circular, placing the hangers at equal distances, etc. [ABSTRACT FROM AUTHOR]

This paper presents a generalized formulation for optimizing the design of concrete beam reinforced with glass fiber reinforced polymer bar. The optimization method is formulated to find the design variables leading to the minimum weight of concrete beam with constraints imposed based on ACI code provisions. A simple genetic algorithm is utilized to solve the optimization task. The weights of concrete and glass fiber reinforced polymer bar are included in the formulation of the objective function. The ultimate limit states and the serviceability limit states are included in formulation of constraints. The results of illustrated example demonstrate the efficiency of the proposed method to reduce the weight of beam as well as to satisfy the above requirement. The application of the optimization based on the most economical design concept have led to significant savings in the amount of the component materials to be used in comparison to classical design solutions. [ABSTRACT FROM AUTHOR]

The aim of this study is to use stochastic optimization algorithm MSAA (Modified Simulated Annealing Algorithm) for trusses plane optimization (weight minimization) with discrete variables. MSAA is based on the cooling process of metal used in the Simulated Annealing (SA) classic, but it has three fundamental characteristics (preliminary exploration, search step and acceptance probability) that differentiate this. To evaluate and validate the MSAA performance were studied three problems plane trusses weight minimization with discrete variables reported in the literature and the results are compared with those obtained by other authors using different optimization algorithms. It is concluded that the MSAA algorithm presented in this study can be effectively used in the weight minimization of truss structures. [ABSTRACT FROM AUTHOR]

This paper deals with the weight minimization of planar steel trusses by adopting a differential evolution-based algorithm. Square hollow sections are considered. The design optimization refers to size, shape and topology. The design variables are represented by the geometrical dimensions of the cross sections of the different components of the truss, directly involving the size of the structure, and by some geometrical parameters affecting the outer shape of the truss. The topology is included in the optimization search in a particular way, since the designer at different runs of the algorithm can change the number of bays keeping constant the total length of the truss, to successively choose the best optimal solution. The minimum weight optimum design is posed as a single-objective optimization problem subject to constraints formulated in accordance with the current Eurocode 3. The optimal solution is obtained by a Differential Evolutionary (DE) algorithm. In the DE algorithm, a particular combination of mutation and crossover operators is adopted in order to achieve the best solutions and a specific way for dealing with constraints is introduced. The effectiveness of the proposed approach is shown with reference to two case-studies. The analysis results prove the versatility of the optimizer algorithm with regard to the three optimization categories of sizing, shape, topology as well as its high computational performances and its efficacy for practical applications. In particular useful practical indications concerning the geometrical dimensions of the various involved structural elements can be deduced by the optimal solutions: in a truss girder the cross section of the top chord should be bigger than the one of the bottom chord as well as diagonals should be characterized by smaller cross sections with respect to the top and bottom chords in order to simultaneously optimize the weight and ensure an optimal structural behaviour. [ABSTRACT FROM AUTHOR]

As the complexities of automotive systems increase, designing a system is a difficult task that cannot be done manually. In this paper, we focus on wire routing and wire sizing for weight minimization to deal with more and more connections between devices in automotive systems. The wire routing problem is formulated as a minimal Steiner tree problem with capacity constraints, and the location of a Steiner vertex is selected to add a splice which is used to connect more than two wires. We modify the Kou-Markowsky-Berman algorithm to efficiently construct Steiner trees and propose an integer linear programming (ILP) formulation to relocate Steiner vertices and satisfy capacity constraints. The ILP formulation is relaxed to a linear programming (LP) formulation which has the same optimal objective and can be solved more efficiently. Besides wire routing, wire sizing is also performed to satisfy resistance constraints and minimize the total wiring weight. To the best of our knowledge, this is the first work in the literature to formulate the automotive routing problem as a minimal Steiner tree problem with capacity constraints and perform wire routing and wire sizing for weight minimization. An industrial case study shows the effectiveness and efficiency of our algorithm which provides an efficient, flexible, and scalable approach for the design optimization of automotive systems. [ABSTRACT FROM AUTHOR]

The aim of this study is to present a new efficient optimization algorithm called Teaching-Learning-Based Optimization (TLBO). The TLBO algorithm is based on the effect of the influence of a teacher on the output of learners in a class. Several benchmark problem related truss structures with discrete design variables are used to show the efficiency of the TLBO algorithm and the results are compared with those reported in the literature. It is concluded that the TLBO algorithm presented in this study can be effectively used in the weight minimization of truss structures. [ABSTRACT FROM AUTHOR]

Abstract: The use of lightweight aluminium sandwiches in the shipbuilding industry represents an attractive and interesting solution to the increasing environmental demands. The aim of this paper was the comparison of static and low-velocity impact response of two aluminium sandwich typologies: foam and honeycomb sandwiches. The parameters which influence the static and dynamic response of the investigated aluminium sandwiches and their capacity of energy absorption were analysed. Quasi – static indentation tests were carried out and the effect of indenter shape has been investigated. The indentation resistance depends on the nose geometry and is strongly influenced by the cell diameter and by the skin – core adhesion for the honeycomb and aluminium foam sandwich panels, respectively. The static bending tests, performed at different support span distances on sandwich panels with the same nominal size, produced various collapse modes and simplified theoretical models were applied to explain the observed collapse modes. The capacity of energy dissipation under bending loading is affected by the collapse mechanism and also by the face-core bonding and the cell size for foam and honeycomb panels, respectively. A series of low-velocity impact tests were, also, carried out and a different collapse mechanism was observed for the two typologies of aluminium sandwiches: the collapse of honeycomb sandwiches occurred for the buckling of the cells and is strongly influenced by the cell size, whereas the aluminium foam sandwiches collapsed for the foam crushing and their energy absorbing capacity depends by the foam quality. It is assumed that a metal foam has good quality if it has many cells of similar size without relevant defects. A clear influence of cell size distribution and morphological parameters on foam properties has not yet been established because it has not yet been possible to control these parameters in foam making. The impact response of the honeycomb and foam sandwiches was investigated using a theoretical approach, based on the energy balance model and the model parameters were obtained by the tomographic analyses of the impacted panels. The present study is a step towards the application of aluminium sandwich structures in the shipbuilding. [Copyright &y& Elsevier]

In structural optimization, most successful sequential approximate optimization (SAO) algorithms solve a sequence of strictly convex subproblems using the dual of Falk. Previously, we have shown that, under certain conditions, a nonconvex nonlinear (sub)problem may also be solved using the Falk dual. In particular, we have demonstrated this for two nonconvex examples of approximate subproblems that arise in popular and important structural optimization problems. The first is used in the SAO solution of the weight minimization problem, while the topology optimization problem that results from volumetric penalization gives rise to the other. In both cases, the nonconvex subproblems arise naturally in the consideration of the physical problems, so it seems counter productive to discard them in favor of using standard, but less well-suited, strictly convex approximations. Though we have not required that strictly convex transformations exist for these problems in order that they may be solved via a dual approach, we have noted that both of these examples can indeed be transformed into strictly convex forms. In this paper we present both the nonconvex weight minimization problem and the nonconvex topology optimization problem with volumetric penalization as instructive numerical examples to help motivate the use of nonconvex approximations as subproblems in SAO. We then explore the link between convex transformability and the salient criteria which make nonconvex problems amenable to solution via the Falk dual, and we assess the effect of the transformation on the dual problem. However, we consider only a restricted class of problems, namely separable problems that are at least C continuous, and a restricted class of transformations: those in which the functions that represent the mapping are each continuous, monotonic and univariate. [ABSTRACT FROM AUTHOR]

Abstract: Morphing Evolutionary Structural Optimization (MESO) is a soft-kill version of Evolutionary Structural Optimization (ESO). This paper proposes a new optimality criteria-based performance index for monitoring the optimization process of the conventional MESO method. Also, a quantitative verification of the MESO for some benchmark problems with equality constraints is presented. For some structures like Michell trusses, a qualitative verification of the ESO method has been already reflected in the literature. However, in this study, a quantitative verification of the MESO method in the shape optimization is shown by comparing the results with analytical solutions. An excellent convergence of the MESO method in stiffness, frequency and weight optimization problems with equality constraints to optimum clearly reveals that this method is a robust optimization technique, which can be easily implemented and applied to a wide range of problems. [Copyright &y& Elsevier]

Abstract: A practical optimization method is applied to design nonlinear truss-like structures subjected to seismic excitation. To achieve minimum weight design, inefficient material is gradually shifted from strong parts to weak parts of a structure until a state of uniform deformation prevails. By considering different truss structures, effects of seismic excitation, target ductility and buckling of the compression members on optimum topology are investigated. It is shown that the proposed method could lead to 60% less structural weight compared to optimization methods based on elastic behavior and equivalent static loads, and is efficient at controlling performance parameters under a design earthquake. [Copyright &y& Elsevier]

Abstract: This paper presents an integrated design procedure for determination of structural arrangement and scantlings for the complete structure of small high-speed craft. The purpose of the procedure is to serve as a tool in the preliminary design stage where it enables generation of weight minimized designs with very limited effort. The design procedure is applied in a material concept study for a high-speed patrol craft. The various concepts include single skin and sandwich composites, aluminum and steel. It is demonstrated that the mass of the aluminum hull structure can be reduced from the original 11.7 tonnes to 9.6 tonnes through application of the presented design procedure. The most weight efficient material concept is a carbon-fiber foam-cored sandwich with a structural mass of 4.8 tonnes, which is about 50% less than the refined aluminum version. Through simple hydromechanic analysis, potential for fuel and CO2 emission reductions of 8% for the refined aluminum version and 27% for the carbon-fiber sandwich version in relation to the original craft are indicated. [Copyright &y& Elsevier]

An approach is presented to determine the design that minimizes the weight in consideration of the postbuckling behavior of stiffened composite panels. Since, due to computational expense, for optimization purposes a nonlinear finite element approach is not appropriate, a semianalytical approach is implemented in the EADS MAS software LAGRANGE. A gradient-based optimization scheme is employed to determine the ply thicknesses and fiber orientation of the composite skins and the cross-sectional areas of the stiffeners. Strength and stability constraints are used. Based on two examples of different complexity, weight savings are shown to be due to the fact, that skin buckling is allowed above a certain load level. [ABSTRACT FROM AUTHOR]

A simultaneous optimization methodology is proposed considering both the structural and the process requirements for the weight minimization of laminated plates manufactured by the Resin Transfer Molding (RTM) process. The coupling between the structural performance and the process cost is considered through the investigation of state parameters in the RTM process. As the structural constraint, maximum allowable displacement or maximum allowable failure index of the structure under load is assigned. For the process constraint, maximum allowable mold filling time is given. A new approach to deal with design constraints without having to concern about the penalty parameters is suggested. The influence of design constraints on the optimal design configuration is investigated through parametric studies. As the optimization scheme, Genetic Algorithm (GA) is employed. To take into account the layer number variation, genetic operators such as the crossover and the mutation are modified. [ABSTRACT FROM PUBLISHER]

In the present work, a procedure for preliminary design optimization of a steam generator has been developed with the objective of minimizing the weight of the generator. Some real engineering constraints are considered in the problem formulation. A method of evaluating the objective function and constraints of the problem is presented. The problem has been solved numerically by using the exterior SUMT in which the Powell unconstrained minimization technique improved by Sargent with the parabolic interpolation method of one-dimensional minimization, is employed. The results of the optimum design and a sensitivity analysis conducted about the optimum point have been reported. [Copyright &y& Elsevier]