Structural model adjustment using iterative methods

Transcription

1 Materials and Structures / Matériaux et Constructions, Vol. 36, November 2003, pp Structural model adjustment using iterative methods N. Roitman, C. Magluta and R. S. Oliveira COPPE/UFRJ Prog de Eng. Civil, CP 68506, Rio de Janeiro, RJ, Brasil ABSTRACT This paper reports a procedure for adjusting structural theoretical models to experimental results based on optimization techniques. It also covers some of the basic concepts of correlation techniques that are normally used to correlate natural frequencies and vibration modes. The developed method is tested performing some numerical simulations with imposed anomalies and through dynamic experimental tests in a simple supported beam. The obtained results show it can be an excellent tool for adjusting structural models and possibly for damage detection using the first vibration modes. RÉSUMÉ Cet article présente un procédé basé sur des techniques d optimisation pour l ajustement des modèles théoriques de calcul de structures vis-à-vis des résultats obtenus expérimentalement. Certains concepts de base sur les techniques de corrélation qui sont normalement utilisés pour mettre en rapport les fréquences propres et les modes de vibration y sont aussi présentés. La méthode développée fut testée, soit par le moyen de simulations numériques avec des anomalies imposées, soit par le moyen d analyse expérimentale dynamique sur des poutres sur deux appuis simples. Les résultats ont indiqué que la méthode ci-présentée peut être un outil très profitable pour l ajustement des modèles de structures, son utilisation pour la détection de l endommagement étant aussi envisageable à l aide des premiers modes de vibration. 1. INTRODUCTION The main objective of structural identification is to adjust theoretical and/or numerical models to experimental results, allowing the development of a numerical/theoretical model capable of representing the actual behavior of a structure. In technical literature, numerical model adjustment and damage detection are normally treated as different matters. This differentiation is due more to the objectives of the analysis rather than the methods actually involved. In both cases, as much as in the structural identification as in damage detection, the understanding of structural behavior is essential. There is a set of methods used in structural mechanics that deal with adjustment of numerical models and damage detection, which have been developed over the last two decades [1-5]. The main difficulty in these methods is the difference in terms of number of degrees of freedom between numerical and experimental models. This is due to the fact that even with an extensive instrumentation, it is very difficult to monitor all degrees of freedom normally considered in the numerical model. The methods for adjusting structural numerical models can be divided into two distinct types: direct and iterative methods. Direct methods express the error between the numerical and physical models (experimental results) through an equation that combines the natural frequencies and vibration modes obtained in experimental tests, and the mass and stiffness matrices of the numerical model. Lagrange operators can be used to minimize this error, leading to a series of matrix operations. Despite the low computational cost of this approach, the corrections carried out on the mass and stiffness matrices of the numerical model do not take into account the physical or geometric characteristics of the structural elements. Therefore it might lead to a model that is adjusted to the experimental results but not necessarily related to the actual structure. On the other hand, iterative methods allow the user to introduce other restrictions, respecting the geometric and physical characteristics of the structural elements and seeking to minimize the differences between experimental and numerical results through iterative algorithms. Another advantage of these methods in relation to direct methods is that they provide a set of parameters that can be used to control the adjustment of the numerical model. These parameters could be chosen, for example, through the user s knowledge of structural behavior or using the correlation techniques between numerical and experimental results. The degree of importance of each parameter in the problem solution could be changed using weighting functions. Logically, the computational cost of the iterative methods is by far greater than that of the direct methods /03 RILEM 570

2 Materials and Structures / Matériaux et Constructions, Vol. 36, November 2003 This paper proposes an iterative method for model adjustment in which a multi-objective optimization algorithm called Goal Programming [6] is employed. This approach turns possible to identify structural damages and/or incorrect hypotheses adopted in the modeling. To illustrate the performance of this method, the analysis of a steel simple supported beam is presented. 2. MODEL ADJUSTMENT METHOD The proposed methodology is based on the minimization of the total quadratic error between Frequency Response Functions (FRF) obtained numerically and experimentally. This error can be written as: N o Ni TOTAL E E (1) p1 q1 where, E f * e j.e j f f f i p - degree of freedom where the response is measured; N o number of measured responses; q - degree of freedom where excitation is imposed N i number of measured excitations; e e * EXP NUM jh jh j EXP NUM j H * j H * j EXP H = Experimental FRF the numerical models. In this program the numerical FRF is obtained with the finite element method using spatial beam elements. The physical and geometric variables used in the numerical model are adjusted in such a way that the error is minimum. This set of variables can be chosen and controlled by the user, establishing their limits and constraints. Fig. 1 shows the scheme of the proposed method. Two distinct stages can be seen in the flowchart. The first one tries to find the minimum error through the Goal Programming algorithm, defining the values for the variables. In the second stage the FRF is calculated using finite elements and the variables from the previous stage In the conventional formulations of linear programming, the objective function has to be written in explicit form. However, in the Goal Programming optimization technique this is not necessary. The variables of the problem describe the physical and geometric characteristics of the elements of the structure. For a better understanding they have been divided into the global and element variables. The global variables are listed below: General correlation of the stiffness matrix through a multiplication factor: this variable performs a general and uniform adjustment of the stiffness matrix; General correlation of the mass matrix by means of a multiplication factor: this variable plays the same role as the previous one, acting over the mass matrix; Concentrated masses: these variables change the concentrated masses in the model to be adjusted; Damping factors: apart from being strictly experimental data, these variables are normally determined with a certain amount of imprecision. The introduction of this variable reduces the degree of imprecision. NUM H = Numerical FRF ( )* - denotes the complex conjugate. The minimization of this total error is carried out by the Goal Programming optimization technique. The Goal Programming technique, developed and presented by Ignizio [6], enables the search for multiple goals under rigid and flexible constraints. In virtue of the additional flexibility of the formulation, a series of simultaneous criteria can be approximately satisfied, with each one being measured by its respective performance index. Appendix A presents a concise description explanation of the Goal Programming technique. A computer program written in FORTRAN language was developed in order to identify anomalies in Fig. 1 - Proposed method. 571

3 Roitman, Magluta, Oliveira The element variables are responsible for the alteration of any physical or geometric characteristics of one or more elements. In the developed system, two types of elements were implemented, the spatial frame and the spring element. Some examples of these variables are: Elastic Modulus Poisson s Coefficient Cross section area Inertia moments in x, y and z directions Density Stiffness of the spring elements. An additional facility of this method stems from the fact that the constraints and objectives are both treated in the same way, although with distinct priorities, giving the user more flexibility to find the targets. Some examples of constraints that have been used are: The lower and upper limits for the element characteristics: these are not difficult to define, as material characteristics are normally well known. The lower and upper limits of the damping factors: the values of the damping rates obtained experimentally by simplified methods can be used, with some degree of uncertainty. In cases where experimental values are not available, a value close to zero can be taken as a lower limit and a value close to one as an upper limit. The lower and upper limits for natural frequencies: these limits can be easily defined by a visual analysis of the FRF s involved. Another possibility is to obtain these limits numerically. And the objectives are: Adjustment of the natural frequencies. A question that was given little attention in this study, and that could be significant for structures with a large number of bars, is that the first numerical natural frequencies are directly compared to the first experimental natural frequencies without the preoccupation that the numerical results could have local modes. Therefore, a technique for the identification of these local modes will necessarily need to be implemented. For example, this can be done through the utilization of the MAC [7]. Minimization of the total quadratic error between the numerical and the experimental FRF (Equation (1)). 3. NUMERICAL SIMULATION To illustrate the performance and capabilities of the proposed method, the analysis of a simple supported beam is presented. The main task here is to identify and localize imposed anomalies. The discretization consists of 10 (ten) elements and the supports are defined by spring elements, with stiffness only in the y direction, as shown in Fig. 2. The material properties and geometric characteristics are shown in Table 1. Fig. 2 - Finite element model of a simple supported beam. Table 1 Properties of the simple supported beam Material Steel Elastic Modulus 2.1 x N/m 2 Poisson s Coefficient 0.3 Density 7.85 x 10 3 kg/m 3 Cross Section Area (A) 6.05 x 10-4 m 2 Inertia moment in x direction (Ix) 5.01 x 10-3 m 4 Inertia moment in y direction (Iy) 2.93 x 10-7 m 4 Inertia moment in z direction (Iz) 3.18 x 10-9 m 2 Length Spring stiffness in y direction (nodes 1 and 11) 1.47 m 1.0 x N/m Three situations were analyzed: (i) alteration of the stiffness in one of the springs, (ii) modification of the geometric properties (area and moment of inertia) of one of the elements of the beam, and (iii) introduction of rotational masses in the supports. In this paper only the last situation is shown, in which the two rotational masses at the extremities of the model are Mz = 0.04 kg.m. An excitation force was applied at node 3 and responses were obtained at four points (nodes 3, 5, 7 and 9), as is shown in Fig. 3. Fig. 3 - Localization of the excitation force and sample points. Table 2 shows the comparison between natural frequencies obtained with the numerical model and reference results. These reference results were obtained numerically, imposing the anomaly previously described. Table 2 Comparison between natural frequencies (Hz) Numerical Result Reference Result F F F Significant differences between natural frequencies can be observed, mainly those associated with the third vibration mode. Such differences indicate that either the quantity of mass was underestimated and/or the global stiffness was overestimated in the numerical model. These differences also can be noted in Fig. 4, where the comparison between the FRF (node 3) obtained for the two models is presented. Clearly, it can be seen that there is also a great discrepancy in terms of amplitude. 572

4 Materials and Structures / Matériaux et Constructions, Vol. 36, November 2003 Fig. 4 Comparison of FRF at node 3 (see Fig. 3). The main discrepancies between the two models could also be identified using Modal Assurance Criteria (MAC) and Correlation Modal Assurance Criteria (COMAC) [7] correlation techniques. The results furnished by these techniques (see Tables 3 and 4) show that there are significant discrepancies between the two models. The last element of the MAC main diagonal has a value lower than 0.9 [8], indicating that the main discrepancy is associated to the third vibration mode. On the other hand, the COMAC analysis shows that the discrepancies occur near the extremities of the beam (see Fig. 3). adjustment obtained with the proposed method. This becomes even more evident when the models are compared before (Fig. 4) and after adjustment (Fig. 5) Table 3 MAC Correlation MAC Table 4 COMAC Correlation COMAC Fig. 5 - Comparison of FRF at node 3 (see Fig 3). Table 6 Comparison between natural frequencies (Hz) Numerical Result Reference Result f f f node node node node From the above results the decision variables were chosen: the areas and the inertia moments for the elements 1, 2, 9 and 10 (Fig. 2) and the rotational masses at the extremities of the beam (nodes 1 and 11). Table 5 shows for each variable its initial value, the initial increment and the results obtained after 50 iterations. It can be seen in Table 5 that the program did not change the areas and inertias of the elements. For the rotational masses the output values were close to the imposed ones (Mz = 0.04 kg.m). This indicates that the program can find the correct values, despite the fact that other variables, not related to the actual problem have been used. This gives the user more flexibility to choose the variables. Table 6 and Fig. 5 show the comparison between natural frequencies and FRF for the two models, using the results obtained in the analysis. These results show the excellent It should be pointed out that forcing the values of the masses to be the same, as a new restriction, the program correctly found the imposed masses in this situation. 4. EXPERIMENTAL TESTS In order to test the developed program in a real situation, some experimental tests were performed on a steel simple supported beam with the same characteristics as those used in the numerical simulation. The experimental results were utilized to adjust the numerical model of the beam. The instrumentation was carried out with four piezoelectric accelerometers to measure the responses, and an instrumented hammer with a piezoelectric load-cell to measure the excitation force, as illustrated in Photo 1. The experimental FRF s were estimated with the H 1 technique, through a system developed in the Structures Laboratory [9]. The modeled beam was the same described in Fig. 2. The comparison between the experimental and numerical natural frequencies is shown in Table

5 Roitman, Magluta, Oliveira Photo 1 - Experimental assembly. Table 7 - Comparison between natural frequencies (Hz) Numerical Experimental f f f In this table it can be noted that the experimental natural frequencies present values below the numerical results. This fact could lead us to the conclusion that the numerical model has a lack of mass or that it is too stiffened. Based on the visual inspection of the structure, it can be observed that the support conditions represent the main problem to the modeling, as seen in Photo 2. Thus, the rotational masses (Mz) and rotational stiffness (Rz 1 and Rz 11) were chosen as decision variables of the two supports. Additionally, the masses of the five accelerometers (m1-5) and the elasticity modules of the bar were adopted as variables. The last one is represented by a constant multiplication of the stiffness matrix (Rk). Photo 2 - Beam support condition. From the comparison between the FRFs it can be concluded that, apart from the existing differences between natural frequencies, the damping rates and/or the amplitudes of the vibration modes of the two models are also different, mainly those concerning the first and second vibration modes. This can be seen through the difference between the peaks of the two models. The direct comparison between the vibration modes showed very slight discrepancies between the experimental and the numerical models. This could also be observed with the MAC and COMAC correlation techniques, since the values found on the main diagonal of the MAC and those of the COMAC are very close to 1.0. Thus, the differences between the peaks of the experimental and numerical FRFs should be associated with the estimates for the damping. Based on these conclusions the damping rates (1-3) also should be chosen as a decision variable. Table 8 shows for each variable its initial value, the initial increment and the results obtained after 67 iterations of the developed program. Through the analysis of Table 8 it can be verified that the variable Rk, which represents the alteration of the steel beam elasticity modulus, did not present any significant variation. This corroborates the idea that the main discrepancy between the numerical and experimental models is associated mainly to the supports of the model. This result also shows that the excessive use of variables does not represent a problem when adjusting the model, since the value of Rk remains unchanged after many iterations. The rotational masses located at the extremities of the model had their initial value estimated from the geometry of the actual support and were considered as being equal. The result showed in Table 8 is very close to the initial value, showing that the estimated value was appropriate. It should be pointed out that this parameter was relevant for the adjustment of the natural frequencies. Although the masses of the accelerometers are small, it was important to introduce theirs weights in order to adjust the numerical model. The value of each mass was estimated by weighing the accelerometer including part of their connecting Comparisons between the experimental and numerical FRF s were carried out. In Fig. 6, a typical FRF is shown, corresponding to the response of the accelerometer installed at node 3 and an excitation at the same point (see Fig. 3). It is worth to highlight that the damping rates used were obtained experimentally through the Orthogonal Polynomial technique [10]. Fig. 6 - Comparison between numerical and experimental FRFs, at node 3, with an excitation at the same point. 574

6 Materials and Structures / Matériaux et Constructions, Vol. 36, November 2003 cables. Although a slight variation could be noted, all of them presented similar values, as it was expected. The rotational stiffness of the supports (Rz) was introduced as a variable, as the boundary condition of the beam may not represent the pinned support properly. As each support could have been assembled in a different way, as for example, the tightness of the screws that connect each of the supports to the steel bar, the rotational stiffness at each extremity were considered as two distinct variables. The obtained results showed different values for these two variables, indicating that one of the extremities is behaving more like a pinned support condition than the other. The initial values of the damping rates were obtained experimentally. The obtained results were fairly coherent with the expected values, i.e., presenting an increase over the damping rates of the two first modes of the numerical model. This can be observed comparing the peaks of the two models shown in Fig. 6. Table 9 shows the comparison between natural frequencies obtained experimentally and numerically, using the results shown in Table 8. Table 9 - Natural frequencies (Hz) Numerical Experimental f f f The adjustment of the natural frequencies can be considered excellent, although the comparison relative to the third vibration mode still shows a small discrepancy between their values. Fig. 7 shows a typical comparison obtained between the regenerated and the experimental FRF. In this figure, the FRFs are relative to the accelerometer installed at node 3 and the excitation at the same point. The comparison between experimental and numerical FRFs shows that the adjustment of the numerical model using the developed method was efficient. This can be noted comparing the FRF peaks in Fig. 7. Fig. 7 - Comparison between numerical and experimental FRFs, at node 3, with an excitation at the same point. 5. FINAL COMMENTS The numerical simulations and the experimental tests indicate that the developed method can be an excellent tool for adjusting structural models and possibly for damage detection using the first vibration modes. The optimization process does not lead necessarily to a unique solution. This is due to the fact that the process could be locked in a local minimum or to the definition of some aspects that depend on the experience of the user, e.g., goals priorities, starting point of the process, initial increment of each variable, number of variables, etc. To avoid the local minimum, it is necessary to execute the system with several different initial conditions. On the other hand, among the aspects that depend on the user, the main aspect is the correct identification of all parameters that should be considered in the analysis. An advantage showed by the proposed method is that the excessive use of the decision variables does not represent a problem for the final adjusting of the model. This fact results in a certain flexibility in the choice of the parameters, favoring the use of iterative methods. The structural model adjustment methods have been extensively applied to mechanical engineering, and more recently to civil structures such as: long span bridges, high buildings and TV towers. For the moment the developed method is been tested for damage detection in fixed offshore platforms. REFERENCES [1] Friswell, M.I. and Mottershead, J.E., Finite Element Model Updating in Structural Dynamics, 1 st ed. Dordrecht (Kluwer Academic Publishers, Netherlands, 1995). [2] Rad, S.Z., Methods for updating numerical models in structural dynamics, Ph.D. dissertation, Imperial College of Science, Technology and Medicine, London, [3] Zhang, O., Zerva, A. and Zhang, D.W., Stiffness matrix adjustment using incomplete measured modes, AIAA Journal 35 (5) (1997) [4] Kim, H.M. and Bartkowicz, T.J., Damage detection and health monitoring of large space structures, Journal of Sound and Vibration 27 (7) (1993) [5] Berman, A. and Nagy, E.J., Improvement of a large analytical model using test data, AIAA Journal 21 (8) (1983) [6] Ignizio, J.P., Goal Programming and Extensions (Lexington Books, 1976). [7] Roitman, N. and Viero, P.F., Identification of damages in offshore platforms: an application of some methods using eigenvectors, Proceedings of the 15th International Modal Analysis Conference, vol. 1, Orlando, 1997, [8] Ewins, D.J., Modal Testing: Theory and Practice (John Wiley and Sons Inc., New York, 1984). [9] Andrade, R.F.M., Development of a system for the experimental determination of frequency response functions using single and multiple excitation, D. Sc. Thesis, COPPE/UFRJ, Rio de Janeiro, 1997 (in Portuguese). [10] Richardson, M.H. and Formenti, D.L., Parameter estimation from frequency response measurements using rational fraction polynomials, Proceedings of the 1st International Modal Analysis Conference, vol. 1, Orlando, 1982, [11] El-Sayed, M.E.M., Ridgely, B.J. and Sandgren, E., Nonlinear structural optimization using goal programming, Computer and Structures 32 (1) (1989) [12] El-Sayed, M.E.M. and Jang, T.S., Structural optimization using unconstrained nonlinear goal programming algorithm, Computer and Structures 52 (4) (1994) [13] Schniederjans, M.J., Linear Goal Programming (Petrocelli Books, NJ, 1984). [14] Powell, M.J.D., An efficient method for finding the minimum of a function of several variables without calculating derivatives, Computer Journal 7 (1964) [15] Vanderplaats, G.N., Numerical Optimization Techniques for Engineering Design: with Applications (McGraw-Hill, New York, 1984). 575

7 Roitman, Magluta, Oliveira APPENDIX A.1 GOAL PROGRAMMING Traditional optimization models require the formulation of a single objective function. This characteristic limits the model s applicability to single objective problem situations. While many real situations can be modeled in this form, a multi-objective formulation is necessary to model many problems. Some problem situations involve not only multiple objectives but also multiple, conflicting objectives. Goal Programming (GP) possesses the capability to solve multi-objective and multi-conflicting-objective problems. A more detailed discussion of GP is given in [6]. In the following, the terminology, basic elements and formulation of GP problems are discussed. A Terminology The following are definitions of several commonly used GP terms and expressions. Decision variables. A set of unknowns (represented in the GP model as x j, where j=1,2,...j) that the program seeks to determine. Right-hand-side variables. Values that usually express resource values (represented by b i ) that one seeks to overutilize or underutilize. Goal. The desire to minimize numerical deviation from a stated right-hand-side value in a selected goal constraint. Goal constraints. A set of constraints that expresses the relevant resource or right-hand-side goals in the problem situation being modeled. Pre-emptive priority factors. A ranking system (represented by P k where k=1,2,...k and K represents the number of goals in the model) that allows goals to be ordinarily structured into the GP model. The ranking system places the importance of goals in accordance with the following relationship: P 1 (most important goal) > P 2 >>>P k (least important goal). Deviational variables. Variables that express the possibility of negative deviation from a goal right-hand-side value (represented in the GP model as d i, where i=1,2,...i is the number of goal constraints in the model) or positive deviation from a right-hand-side value (represented by d i ). Differential weights. Mathematical weights that are expressed as cardinal numbers (represented as w where kl k=1,2,...k; l=1,2,...l) and are used to differentiate the l deviational variables within a single k, priority level. Satisfying. A concept that implies that GP methods seek a solution that fully satisfies as many goals as possible rather than optimizing a single goal. Technology coefficients. Numerical values (represented by a ij ) that express the per unit usage of the b i value in the creation of the x j. A Basic elements There are several basic elements common to any type of GP problem. These elements provide the common framework in which any unique problem situation can be expressed. Objective function. One of these basic elements is the GP objective function. Three types of objective functions exist, and they can be expressed as follows: I Minimize: Zd d (A1) i i i1 I Z P ( d d ) k i i Minimize: i1 (for k=1,2,...k) (A2) I Z w P ( d d ) kl k i i Minimize: i1 (for k=1,2,...k, l=1,2,...l) (A3) In Equation (A1), Z represents the sum of all negative deviation d i and all positive deviation d i in I goal constraints. This type of objective function applies when deviational variables are not distinguished by priority or weighting. Essentially this type of problem seeks to minimize the total deviation from all right-hand-side values. Equation (A2) expresses an objective function where there can be goals ranked by P k priorities. This type of objective function suffices when ranking or ordering of goals is required, but the deviational variables within each priority level are of equal importance. Equation (A3) expresses an objective function for which goals are ranked and deviational variables within each priority level are differentiated by the use in the w kl differential weighting. Goal constraint. A second basic GP element is the goal constraint. Three types of constraints exist, the purpose of each being determined by its relationship with the objective function. Each type of constraint must have one or two deviational variables placed in the objective function. In the following, the basic three constraints types are discussed: ; ij j i i (A4) ; ij j i i (A5) ij j i i i (A6) a x d b a x d b a x d d b Equation (A4) permits negative deviation. Equation (A5) permits positive deviation. Equation (A6) permits deviation in both directions. Non-negativity requirements. A third basic element in the GP method is a non-negativity requirement. All model variables must be greater than or equal to zero. Since the 576

8 Materials and Structures / Matériaux et Constructions, Vol. 36, November 2003 model contains decision and variables, the non-negativity requirement may be stated as: x, d, d j i i 0 (A7) subject to: G j (x)0 (for j=1,2,...,j), H k (x)0, (for k=1,2,...,k) (A9) (A10) A Formulation Formulating an GP problem requires the determination of the decision variables x j, the technological coefficients a ij, and the right-hand-side values b i. There are six basic steps one may take when formulating an GP model [13]. 1. Definition of decision variables. In this step, one clearly states what the unknown decision variables represent. A more precise definition eases the remaining formulation procedure. 2. Formulation of goal constraints. This step requires the identification of the right-hand-side variables and then the determination of the appropriate combinations of technological coefficients and decision variables which will affect the right-hand-side. One should also pay close attention to the types of deviations allowed in each constraint and specify them correctly. 3. Determination of pre-emptive priorities. In this step one should determine if any ranking or ordering of goals is necessary for the problem. If no ranking or ordering is necessary, this step should be skipped. 4. Determination of differential weights. This step requires one to determine what preferences or rankings exist within a specific goal level. If none exists, this step should be skipped. 5. Definition of objective function. The key to this step is to select the correct deviational variables for inclusion in the objective function. In this step one should also attach the appropriate pre-emptive priorities and differential weights to the appropriate deviational variables. 6. Enforcement of non-negativity. This step simply requires one to set all decision variables and deviational variables greater than or equal to zero. This assures that no negative solutions are generated. A.2 - OPTIMIZATION PROBLEM The general optimization problem to be solved presents the following form: Minimize:F(x) T N x=x, x,..., x R (A8) 1 2 n and l u x x x (for i=1,2,...,n) (A11) i i i where F(x) represents the objective function, a scalar function of the decision variables x. G j (x) represents the inequality constraint functions. H k (x) represents the l u equality constraint functions. x and x i i are lower and upper bounds respectively on the decision variables. A.3 NONLINEAR GOAL PROGRAMMING USING POWELL S METHOD There is a variety of methods that can be used to solve nonlinear optimization problems. A straightforward approach is to use a zero-order optimization problem. Powell [14] introduced a method of multi-variable optimization based on the concept of conjugate directions, and this method is one of the efficient and reliable nongradient methods available [15]. The basic concept of Powell s method is first to search in n orthogonal directions, where each search consists of updating the decision variables using the minimum along the previous search direction as the starting point. After performing these successive minimizations, a new search direction is formed between the original starting point and the resulting point of the successive n searches. The first search direction is then dropped and the remaining search directions are kept along with the new direction, which is placed last among the directions. The search is continued until convergence is achieved. The nonlinear Goal Programming algorithm [12] used in this work was developed using Powell s algorithm as the optimization routine. The nonlinear Goal Programming first minimizes, as nearly as possible, the objectives with the highest priority level. It then proceeds to satisfy the objectives of the next priority level. This process is continued until all priority levels have been considered. At each priority level, the search is terminated when the difference between present and previous function value becomes sufficiently small. Paper received: March 19, 2002; Paper accepted: July 2,