AN INTERACTIVE OPTIMIZATION COMPONENT FDR SOLVING PARAMETER ESTIMATION AND POLICY DECISION PROBLEMS IN LARGE DYNAMIC MODELS DIssertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY MARCUS ROBERT BUCHNER 1975 ‘llzllllllllil 2‘ 4 "1 n. This is to certify that the thesis entitled AN INTERACTIVE OPTIMIZATION OOMPONE '7 CFOR SOLVING PARAMETER ESTIMATION AND 'CY DELS DECISION PROBLEMS IN LARGE DYNAMIC presented by Marcus Robert Buchner has been accepted towards fulfillment of the requirements for Ph.D. deyeeinEIectricaI Engineering 8 Systems Science KI Oratzlizli‘ ,J/{Mfi 4 0-7639 I I .2; 3" ’3 svsf 41‘30 7 m" ' '7 995 “ 2"- W) ' ‘ I ”I: 3'0! W 35! d9? «e / ") Aesncr ( A {M AN INTERACTIVE OPTIMIZATION couponm ma SOLVING 9mm ESTIMATION ‘4. AND pone! DECISION momma IN LARGE DYNAMIC norms '\ _ By Marcus Robert Buchner There recently has been an increased use of large. nonlinear. and dynamic models to represent real world phenomena. This primarily results from the digital computer's capability to produce numerical solutions for complex differential and difference equations. However. parameter estimation and policy decision problems resulting frem the use of these models often cannot be analytically solved. This thesis develops an interactive optimisation component. an integrated set of minimisation techniques. to assist in the efficient solution of these problems. The component was developed by reference to an efficiency measure depending on four factors: (1) component costs resulting from the use of the computer. (2) the ease with which interaction with the compo- nent can be achieved. (3) the difference between component solutions and true problem solutions. and (h) compatibility of the component with different computers. In the thesis. particular emphasis was placed upon certain aspects of the efficiency measures the problem solution accuracy. the number of simulation runs necessary to achieve convergence to the solutions. the interactive capability to change search structure and parameters during the component execution. and computer memory and central processor use. A literature search of optimization methods revealed the Marcus Robert Buchner particular advantages and disadvantages of the individual algorithms. In a specific type of problem one method generally performs more effi- ciently than other methods. However. it is very difficult. if not in- possible. to determine a priori which method would be most efficient in a given application. In addition. the literature search of previously developed optimisation components led to the formulation of a set of basic requirements for an efficient component. constructed to be FORTRAN compatible. the component consists of two sections: a pattern recognition sub-component which has the primary capability of examining user specified model behavior. and an optimiza- tion sub-component that has the capability of performing efficient op- timizations to solve parameter estimation and policy decision problems. Each sub-component is built to allow interaction with a general form of simulation model while the FORTRAN compatibility allows execution on a variety of computers. The pattern recognition sub-component allows an examination of model behavior in either a printed or graphical form. In addition. comparisons between model outputs. real world data. and smoothed real world data can be made with the component. Therefore this section of the component can assist in the solution of parameter estimation and policy decision problems in three principle ways; (1) through rapid isolation of regions in the input variable space representing invalid or insensitive behavior. (2) through rapid isolation of regions in the variable space needing detailed investigation. and (3) through assist- ance in the determination of the validity of the model. The optimization sub-component contains an efficient two-level interactive optimzation algorithm. In the algorithm. the Complex Marcus Robert Buchner method and Powell's method are integrated into a search technique that is effective both far from and in thervicinity of a local optimum. In addition. methods have been developed to insure compatibility between the search methods. In other words. information gained from using the Complex method is incorporated in the initialization of Powell's method! thus the convergence rate of the component is increased. The test results demonstrate that the component is an effective tool for solving parameter estimation and policy decision problems; the component results in improved solution accuracy and increased speed of convergence to the solutions in both problems. In particulars (1) When solving a parameter estimation problem with the component. the use of the smoothing capability to filter real world data improves the problem solution accuracy. This occurs when the data result from a process corrupted with zero-mean uncorrelated noise. (2) The addition of techniques to insure the compatibility between the Complex and Powell's methods increases the eff1c1ency of the compo- nent when the objective function is smooth and well behaved. (3) The Complex-Powell two-level search algorithm generally reduces the number of simulation runs necessary to find good parameter estimates or controls as compared to either method used separately. (h) The modification to Powell's method to speed convergence along sharp ridges and valleys also results in a reduction of simulation runs necessary to obtain solutions. (5) Throughout the testing phase. the component's interactive capability provided for a significant improvement in the speed with which the component converged to problem solutions. There are five principle new developments of this thesis that Marcus Robert Buchner contribute to research in the solution of parameter estimation and policy decision problems in complex simulation models: (1) the incorpo- ration of data filtering techniques to improve the solution accuracy of dynamic. nonlinear. parameter estimation problems. (2) the development of an integrated. interactive. general optimization component. (3) the development of a modification to Powell's method to improve the rate of convergence on poorly behaved functions. (u) the derivation of a tech- nique to approximate the variance of parameter estimates in complex estimation problems. and (5) the design of a two-level search algorithm to efficiently locate local optima of a wide variety of objective func- tions. The optimization component can assist in the solution of problems often encountered in a systems approach such as model validation. model tuning. and model control. In the model validation and model tuning stages of development of a model. the component can provide parameter estimates that result in minimizing an error norm between real world and model behavior. Also the calculation of the sensitivities and variance of the parameter estimates by the component provide valuable information useful in the validation and tuning stages of model devel- opment. In the area of model control. the component provides approx- imations to control functions that realise specific desired behavior of the model. The research for this thesis revealed several areas needing further investigation. Some of these areas include the design of an efficient decision algorithm for automatic transfer from the Complex method to Powell's method. development of techniques to cope with autocorrelated noise in the real world time series and strong interaction in the Marcus Robert Buchner search variable sets. and further development in the selection of appropriate basis sets for representing controls in a policy decision problem. In summary. although there are many areas connected with the research for this thesis that need further investigation. the optimiza- tion component can provide good problem solutions for certain classes of parameter estimation and policy decision problems. AN INTERACTIVE OPTIMIZATION COMPONENT FDR SOLVING PARAl‘IETER ESTIMATION AND POLICY DECISION PROBLEI‘B IN LARGE DYNAMIC MODELS By Marcus Robert Buchner A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of mm OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1975 Copyright by MARCUS ROBERT BUCHNER 1975 This work is dedicated to my father who instilled in me a desire to know. ii ACKNOWLEIDD‘IENTS Without the help of many people I could not have realised the completion of this thesis. There are some . I feel. that deserve special recognition for their guidance and help. First. I would like to thank my major advisor. 11‘. Thomas Manetsch. who has provided direc- tion and assistance for my research. and has spent mam hours helping me to prepare this manuscript. Dr. Manetsch has been of incense help in aw professional development and for this I am extruely grateful to him. The other members of my graduate comitte- 11‘. Robert Barr. Ir. Leroy Kelly. and 11-. Gerald Park- have provided guidance for aw graduate education and this thesis. Their help has been in many diverse areas ranging from my personal to my professional growth. Next. I want to thank Chris Wolf and Keith Olson for their programing assistance. Finally. and most importantly. I would like to acknowledge aw wife. Ellen. and express m extreme appreciation and thanks for her help in ' the preparation of this manuscript. for her financial and moral support during my education. and for her personal strength allowing me to com- plete the research for this thesis. In addition at the above acknowledgements. I want to also express my appreciation to the Department of Electrical Engineering and Systems Science and AID contract/csd-2975 for financial support of this research. Without this aid. this thesis could not have been completed . iii TABLE OF (DNTENTS LIST OF TABLES LIST OF FIGURfi INTROWCTION CRAFTS! I - GENERAL CONCEPTS General Backggund Parameter Estimation Randomness Policy Decision Making CHAPTHI II - PROBLDI COBIDERATIONS Problem Concepts Problem Statement Review of Previous Work Optimisation Methods Direct Methods Univariant Search Simplex Method Complex Hooke-Jeeves Search Powell' s Method Rosenbrock's Method Gradient Methods (Modified Davidon's Method) Optimization Components Univariant-Pol omial Approach 8 Tex-Pol omial A ach im Qp imisa ion CAIBIS Cgponent Reguirements CHAPTER III - DESQIIPTION OF THE OPTIMIZATION COMPONENT General Descriptigp Detailed Description Pattern Recognition Sub-Component ProJLam MAINPR Subroutine SUBP§_ Subroutine $1100ng Subroutine PLOT Optimisation Sub-Component Progr_am MAINOPT Subroutine SUmPT Subroutines WW iv vi vii H Subroutine SENS Subroutines BOX, HDTMI and TRANS Subroutines FUNC and TRANSFE Subroutines Sm and 8m Subroutine SPEED Subroutine GLOBE Subroutine COMP Subroutine LINAPP User Supplied Subroutines. SIM. NNN, and MMM CHAPTER IV - TESTING ppgcription of the Test Model Larameter Estimation Test Problems Test 1: Validation of Optimisation Approach in Solving Parameter Estimation Problems. Test 2: Complex-Powell Optimization Algorithm Test 3: Data Smoothing Capability Test at Powell's Speed-Up Algorithm Interaction Capabilities Policy Decision Test Problem CHAPTER V - CONCLUSIOIB AND AREAS 1m PORTER! RESEARCH Conclusions Further Extensions APPENDIX A-Subprogram Cross Reference List APPENDIX B-Var iable Dimensioning APPENDIX C-Component Use of Input-Output Tapes APPENDIX D-Component Interaction Statusents APPENDIX E-Example of Component Use APPENDIX F-Program Listing BIBLIOGRAPHY 59 6O 65 69 7o 71 72 78 83 86 93 100 103 105 11a no 123 126 129 132 135 11.3 150 181 Table Table Table Table Table Table Table Table Table Table Table III-1 III-2 IV-1 III-2 IV-6 IV-7 III—8 LIST OF TABLES Instruction Array Elements and Corresponding Options Adaptation of SIMEX Simulation Component Solutions for r =0. l :1. With and Without Model Mor Comparison of Search Algorithms By Function Evaluations Needed to Achieve Convergence Results of Tests to Determine Approximate Functional Dependence of N(n) Results of Tests With a8=1 bror Criterion of the Smoothing Capability of the Component Using Real World m. 5.1-1... Generated um. 0:12. Uncorrelated Noise. and EBASEgéass Results of Tests With a 3:1 bror Criterion of the Smoothing Capability of the Component Using Real World rm. Sen-1.. enerated With 1:92. Uncorrelated Noise. and EBASETB E Results of Tests With a K20 Error Criterion of the Smoothing Capability of the Cosmonent Using Real World Time Series Generated With 1:12.).=0.5 Autocorrelated Noise. and EBASféASE Results of Optimizations to Test Powell's Speed-Up Modification With the Nigerian Cattle Model Results of Series Representation of Optimal Controls Sumary of Component Testing Results vi 7a 95 101 101 101 101‘ 108 115 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure III-1 III-2 III-3 III-1+ 111-5 III-6 III-7 III-2 III-3 III-4A IV-UB III-4C LIST OF FIGURE-B STRUCTURAL CONNECTIONS TO COMPONENT FLOW CHART OF THE PATTERN RECOGNITION SUB-COMPONENT FLOW CHART OF MAINPR FLOW CHART 0F SIDOTH FIN CHART OF OPTIMIZATION SUB-CMONENT DETAILED F1041 CHARTS OF THE SUEIOUTINES ml. MM. SENS. AND SPEED FLOW CHART OF SUHIOUTINE FUNC Flow Chart of Economic Test Model Graphs of Model and Real World Time Series Graphs of Model and Real World Time Series Graphs of Input and Output Variables vs. Time for a :0 Graphs of Input and Output Variables vs. Time for a=a - -1 Gra he of Inppt and Output Variables vs. Time for 932 and 2:2 vii 1&1 5 53 81 89 111 112 113 INTRODUCTION This thesis describes an optimization component that was developed to assist in the solution of parameter estimation and policy decision problems in complex models. Included in the component are numerical optimization methods. special techniques to deal with problems associ- ated with optimizations. and the capability of interaction with the executing program. Chapter I contains a broad overview of pertinent systems concepts: modelling. simulation. parameter estimation. and policy decision mak- ing. Chapter II provides the mathematical formulation of the problem and definitions of terms used later in the thesis. Also included in the second chapter is a review of research conducted in the areas of optimization methods and component development. This review indicates the problems that had to be solved to develop the component. A de- tailed description of the component and the theory of its deve10pment is contained in Chapter III. Chapter IV contains the results of tests performed on the component in problems of parameter estimation and policy decision making with a complex model. Conclusions and areas of further work are discussed in Chapter V. CHAPTER I GENERAL CONCEPTS General Backggggpg_ Mathematical models are finding increased use as means of design- ing and managing real world systems. The purpose of this thesis is to describe an interactive optimization component to aid in solving param: eter estimation and policy decision problems in complex mathematical models. This chapter provides a broad overview of the subjects of model- ling. simulation. parameter estimation. and policy decision making for the reader not familiar with a systems viewpoint. A reader acquainted with these concepts should immediately proceed to Chapter II. the pro- blem formulation and review of the literature. A model is a description of a system in a language that allows a convenient study of the system’s behavior to be made. Thus. obtaining a model that adequately reflects the behavior of the system in the variables of the model is the goal of model construction. Models that describe relationships between system properties with the use of math- ematical equations are the exclusive concern here. An important question in modelling is whether or not the mathemat- ical model realistically represents the real world. This problem of validation has no definite or universal answer. However. there are at least four commonly accepted criteria for model validation: (1) con- sistency of the model equations with the physical and logical 2 3 constraints of the system. (2) intuitively reasonable behavior of the model as determined from a familiarity with real world systems behav- ior. (3) agreement of model behavior with available data from the real world. and (4) capability of forecasting or predicting new behavior of the system under consideration. If used in the model development, testing and application stages. the optimization component described later provides information helpful in determining whether these cri- teria are met. At apprOpriate points in this thesis. these criteria are discussed in more detail. In using a model to study a systemFs behavior. it is convenient and helpful to solve the model equations. This means obtaining an explicit. analytical relationship between the input. state. and output variables of the model. The "inputs" are the variables influencing the behavior of the model while the “outputs” are those variables express- ing the behavior. The variables that describe the history or memory of the model are called the ”states.” In the past. many models contained simplifications in the mathe- matical equations relating variables so that an analytical solution to the models could be found. This resulted in models that did not ap- proximate the behavior of complex systems very well. Simplicity in the mathematical equations and ease of solution were primary factors in determining the model to be used. rather than the model being de- termined by the criterion of’realism in the model's behavior. With the advent of the digital computer and the capability of per- forming large amounts of numerical computation in short time periods. methods were developed to construct numerical solutions to complex mathematical equations which previously had no known analytical 4 solutions. This innediately led to the construction and use of complex models to represent real world behavior. Complexity in a model arises in three principle ways. First. a model may contain nonlinear relationships between variables. A linear relationship exists between two variable subsets. I and I. if they are related such that y:Ax. where y1=Ax1 . yzzsz. implies ayleby2= anlebe2=AaxleAbx2=A(axlebx2) for any two real numbers a and b. and A is a mapping from the set 1 consisting of all possible x. to the set I consisting of all possible y. If a model is linear. or the relation- ship between the inputs and states. and outputs is linear. then the sum of any two solutions is a solution of the model and any constant times a solution is also a solution. This property of linearity in a model reduces the difficulty in finding an analytical solution. whereas its absence has the opposite effect. Secondly. many models are large in the sense of having many varia- bles and relationships in the model equations. This is to realisti- cally approximate real world behavior and to provide to the user of the model adequate information about the model's behavior. For exam- ple. if a model of a country's agricultural sector is to be developed for use in economic planning. then many variables and variable rela- tionships would have to be taken into account for new different com- modities. regions. and processes. A failure to realistically model the sector might prove disastrous if the model were to be relied upon for constructing the country's agricultural policies. A user of this model would be interested in the effects on new variables. such as prices. demands. and yields of comedities. resulting from changes in policy. Clearly. a large model would be needed for this example. 5 Dynamics is the third property of models that contributes to their complexity. This means that they consist of differential and/or dif— ference equations describing the relationships between variables. In a dynamic model. as opposed to a static one. the outputs depend on the inputs that have occured in the past. In short. the model exhibits characteristics of having memory. For example. a dynamic relationship between x and y. functions of time. might be y(t)= £:16(s)ds. However. y(t)=x(t)+xg(t) is an example of a static or non-dynamic relationship. Given x in both examples. it is clear that it is more difficult to find y in the dynamic example than in the static one. Dynamics in a model also introduces difficulties in finding analytical solutions. One method of constructing solutions to complex models is simula- tion. In a simulation the model solutions are developed numerically over time from.one instant to another. Essentially. a simulation is an approximation of the model equations by another set of equations or a ”model of a model.” As a result of the complexities in models. simulation is nearly always needed to find solutions. In addition. the amount of numerical computation required in many simulations makes the computer necessary for calculations. Although simulation is a significant aid in studying the behavior of models. there are many problems that confront the user of a simula- tion: (1) how to decide whether the numerical solutions provided by the simulation agree with the solutions of the model equations. (2) how to construct a simulation so as to economize computer time and storage. (3) how to estimate the parameters in the simulation. and (4) how to select policies for control of the system using the simu- lation. This thesis addresses the latter two problems and details a 6 method to assist in their solution that can be used with large. non- linear. and dynamic models. Parameter Estimation In all models of systems there are variables that can be consid- ered as either constant or fixed functions of time independent of the inputs to the model. These variables are referred to as parameters. If the equations of a model are given. and the parameter values un- known. then the process of finding parameter values that cause the be- havior of the model to closely approximate the system's behavior is called parameter estimation. For example. if in a model of a human population it was assumed that the female population was a constant fraction of the total population. then the model equation representing this would be FP=3¢TP. where F? is the female population. TP is the total population. and A is a parameter independent of the populations. If in three populations of 1000 individuals it was found that there were 513. 525. and 560 females respectively. then an estimate of 3 might be (-g%g— e -ngg— e -ngg-)l3. This estimate is the average prOportion of females to total population for the three samples. For another example. concerning the investment of money. it might be as- sumed that a certain initial investment. M(O). in the stock market group over time according to the linear differential equation dM(t) :b-M(t). where MKt) is the amount of money that the original investment is worth after t years. Here 8 is an unknown parameter that is to be estimated from data obtained from previous investments in the market. For a third example. consider the model of a country's agricultural sector that was discussed earlier. Parameters that might be included in this model are price elasticities of different 7 commodities. yields of commodities under different conditions. and average time delays of various processes. In this example. some form of data obtained from historical records. experimental information. or sampling must be used to estimate the parameter values that result in a good or satisfactory agreement of the model with the real world. Some factors common to parameter estimation problems are evident from the examples given. Clearly in any parameter estimation problem there must be some measure of how closely the model's outputs match the real world data. In different problems different measures might be appropriate. In the first example of parameter estimation. one might be interested in finding. for all samples available. the value of 2 that minimizes 1EIFPJSF‘TFPiI . where there are M sample pepula- tions. FP1(;) is the model's value of the female population for the i'th sample with parameter value equal to'?. and TFPi is the true. real world. value of the female population in the i'th sample. In the second example. a previous investment might have grown over time ac- cording to the function N(t) with N(0)=Nb. One then might be inter- ested in finding the value of b that minimizes £:4§N(s)eM(B.s))2ds. dM(b:t) : S.M(g.t) With M(B,O)=Noo where M(b.t) is a solution of This would result in the model solution M being close. on the average. to the real world data N over the time interval [0.T]. From these two examples. parameter estimation is seen to be an Optimization problem: or to estimate parameters. it is necessary to maximise or minimize an appropriate measure of fit between the model outputs and the real world data with respect to the parameters. Another factor common to parameter estimation problems is that it is necessary to have information about the behavior of the system 5 under study. One problem might have a different type and quality of information available than another. With a simple electrical system consisting of a resistor and a voltage source. it might be necessary to measure the conductance of the resistor. a parameter of the system. If accurate voltage and current measurements could be obtained under a wide variety of conditions. then this should allow for an accurate and reliable value of the conductance to be estimated from the available data. On the other hand. if information about an economic sector of a country is needed. it mdght be possible to obtain only yearly data over a short time span. In addition. the data available might be subject to errors from.mistakes in recording and inflated or deflated performance figures. Presumably. parameter estimates resulting from this situation would be unreliable. Given the model. data from the real world. and a measure of how closely the model outputs agree with the real world data. the problem in parameter estimation is to find the parameter values that optimise the measure of fit. If'the model realistically represents the real world. the data adequate to allow an optimum of the measure to be found. and the measure chosen appropriately. then the resulting para- meter estimates can be relied upon in the model to study the behavior of the real world. However. as in the case of attempting to solve model equations. the parameter estimation problem eludes an analytical solution when nonlinearities. large number of parameters. and dynamics are present in the problem. The problem of parameter estimation in static linear models has been effectively solved with regression techniques resulting from the 2 GausséMarkov theorem. 7 These techniques are employed in the 9 discipline of econometrics. the empirical determination of economic laws. In addition. econometrics provides methods to estimate paramr eters in discrete dynamic linear models. Although econometric tech- niques yield valuable information about parameter estimates and esti- mator variance. they cannot be generally applied to nonlinear estima- tion problems. Similarly. in dynamic linear systems two types of techniques are used to solve the parameter estimation problem: (1) estimation of the transfor functionb'17 of the model using the model's response to an impulse. sinusoidal signal. or wideband gaussian noise. and (2) direct estimation of the model parameters 1? using techniques comparing model outputs with system outputs such as Kalman filtering. ‘What then can be done to solve these problems in complex models? Because parameter estimation is essentially an optimisation problem. numerical techniques and methods that have been useful in function optimisation can also be used to assist in solving these problems. A numerical optimisation method is an algorithm. or set of well defined instructions. for finding an approximation to the maximum or minimum of a function by evaluating the function. and derivatives if availa- ble. for different arguments. For an example. consider the functiOn. mentioned earlier. F(;)=.§IEP1(;)-TFP1|‘with possible values of 3 ranging from 0.0 to 1.0. One possible optimisation method that can be used here 1. to evaluate F for 2:0.000. .001. .002. .... .998. .999. 1.000. The value of 3 in this set that minimises F can be taken to be an approximation to the best parameter estimate. This method requires only that F be capable of numerical evaluation for given values of its arguments. Because a model can be solved numerically by simulation given 10 values for the model's parameters. inputs. and initial states. any measure of fit between the real world data and the model outputs can be calculated for a given set of parameter values. Thus. an Optimiza- tion method that only involves function evaluation. called a direct method. can be applied to any such well defined parameter estimation problem regardless of the complexity of the model that generated it. However. additional problems. connected with optimisations. arise when using these methods. These problems and their possible solutions are discussed in detail later. The optimisation methods considered here are sequential which means that improved parameter estimates are calculated from previously generated information about the measure of fit. This contrasts to a simultaneous approach where the measure of fit is evaluated at pre- determined values of the arguments. and the best parameter estimates are calculated from.the information generated. An example of e si- multaneous method was discussed in connection with minimising F(3). Although sequential optimisation methods more efficiently use informa- tion about the local nature of the function to be optimised. the si- multaneous methods produce valuable information about the general nature of the function.3 There are four basic operations in sequential methods needed for estimating parameters in complex models: (1) Select an initial guess of the parameter values for the parameters to be estimated in the model. (2) the a simulation run with the most recent parameter esti- mates (or additional runs as required by the particular method) and calculate the measure of fit for the parameter 11 values. (3) Update the best estimate of the parameter values with the information gained in step (2) and previous iterations of the algorithm. (a) If not satisfied with the behavior of the model with the last set of parameter values return to step (2); if satisfied. use the last set of parameter values as the parameter estimates for further experimentation with the model. In attempting to apply an Optimization method to solve a parameter estimation problem of a complex model. many difficulties 2'3'25'29 are encountered: (1) each Optimisation method has different charac- teristics. it being impossible to determine beforehand whether a par- ticular method will be satisfactory in a given problem. (2) errors in the data may result in poor parameter estimates. (3) the model might not be capable of representing reality satisfactorily. and (a) auto- matic estimation methods might yield unreasonable parameter estimates. The Optimization component described in this thesis assists in solving these and other difficulties related to optimization problems. An Optimization component is a computer program that integrates a set of optimization methods. special techniques to deal with the problems associated with Optimizations. and the capability of interaction with the user of the component. The objective of the research that was conducted fer this thesis was the development of a component with these capabilities. Randomness In this thesis the concept of randomness is used in two fundamen- tally distinct ways. First. to model randomness inherent in the 12 behavior Of a system. parameters can be represented as random.varia- bles: or to realistically model certain real world processes it is necessary to introduce randomness in the parameter values. Although the main concentration here is with deterministic models. the require- ments developed fer the Optimization component resulted from an aware- ness that the component also be able to assist in solving parameter estimation problems in stochastic models. Because Of the emphasis on the solution of deterministic parameter estimation problems here. the testing was conducted with a deterministic model. Second. to test the efficiency Of the component to obtain good parameter estimates in the presence of a noise factor in the real world data. understood to have resulted from.an underlying deterministic process. artificial real world data were generated. These sets of data. model outputs corrupted with additive noise components. were used to test the efficiency of the component to find parameter estimates by optimizing a measure Of fit between the model outputs and the real world data. The intended mean- ing of the term randomness. or noise. should be clear from the context. Policy Decision Hiking Although the problem Of policy decision making. or Optimal control. is a different one than that Of parameter estimation. the optimisation component developed here can be used to assist in solving this problem as well. Often it is desired that a system or real world process ex- hibit a particular type of behavior. It may be that certain variables must remain within fixed bounds or that the system is to be in or near a given state after a particular time period. The problem Of realiz- ing specific desired behavior of a system is called the Optimal con- 1 trol or policy decision making problem. ‘with the system model used to 13 examine the effects of various policies or controls. If the model is realistic. then the policies used to realise the desired behavior Of the model can also be used to realise the same. or nearly so. behavior in the real world. FOr an example. consider the relationship between the distance. x. Of an Object with unit mass from a given point. and the acceleration. i} iKt)=u(t). where u is an externally applied force. The problem of how to move the Object from an initial position. x(0). to a target. xf. in the least amount of time with u constrained such that N i U kg- 1%? is an Optimal control problem. For another example consider the model. previously discussed. Of a country's agricultural sector. A policy decision problem might be to find the manner in which to regulate government purchases and sales so as to achieve in- ventory levels and commodity prices within certain ranges while mini- mising government spending. In the area Of policy decision.mmking there are two uses Of an Optimisation components (1) where a performance function can be identified that represents how well or efficiently the model responds to policy decisions. and (2) when there are more than one relevant criterion of perfOrmance Of the model behavior that cannot be incor- porated into a single perfbrmance function. With multiple criteria. examining tradeoffs between different policy decisions is important. In the first case. an Optimisation component can directly assist in the solution Of the problem. In the second case. the component can produce a sequence of constrained Optimisation problem solutions that either displays tradeoffs between constraints and performance indices or converges to a solution that realises the feasible desired goals 1“ and objectives of the system behavior. Classical linear systems control theory.17 essentially linear feedback analysis. has been used to study relatively low-order feed- back control systems. Classical techniques have been found to be adequate to deal with a wide variety Of problems arising in the design and control Of systems. Similarly. Optimal control theory.1 based on the minimum.principle of Pontryagin. and dynamic programming. based on the work by Bellman. have been successfully applied to analytically solve a class of particularly well-defined linear and nonlinear dynam- ic optimisation problems. However the solution of many complex con- trol problems of interest are extremely difficult if not impossible at the present time. A The component described in this thesis can aid in solving control problems fer which there are no known analytical solutions. This is accomplished with the component by Optimising a sequence of perfor- mance functions with respect to the coefficients of a series expansion of the control function. An example of the use Of the component to solve a complex control problem is described in Chapter IV while the fellowing chapter contains a review of the literature and the problem formulation. CHAPTER II PROBLEM CONSIDERATIONS Problem Concepts In order to formulate the problem. it is necessary to first intro- duce several concepts and define certain terms. As discussed in the introduction. the principle concern here is with parameter estimation and policy decision problems in large. nonlinear. and dynamic models. Such a model can be described by a set of differential and/or differ- 1? ence equations: 31m = also). am. 20:). as» {mm = gage). 3(15). gm. 3(t)) 2-1 1(3. 2. t) = you» where x} is a p X 1 vector valued function of time. 5? is a q I 1 vector valued function of time. =91(t) E {(s. :28; ) s s _<_t} is a set of ordered pairs rep- resenting the state Of the model at time t. y'is an r X 1 vector function of time representing the outputs Of the model. 9a. 92. and §_are vector valued functions. 2'is the set of n parameters of the model. b is the set of uncontrollable and possibly stochastic inputs to the model. and 3.1s the set of controllable inputs to the model. The set of model equations (2-1) can be approximated by another set 15 16 that defines a simulation of the model. The first p equations in (2-1) are integrated from t to teAt giving: 1 1 t+At 2S (teAt) = E. (t) *It; §1(,3_(2). 3(2). §(z). 1_1_(s)) dz 2-2 Approximating the integral by ;(§(t). gt). 20c). 2(t). A t) 2-3 where I is an approximation of the integral Of g1. from t to teAt. yields aim At) 51m . _I_(_s_(t). gm. 2m. 30:). A t) 2 a or 5%» At) = g';<_s__ O. j=1. 2. .... where x and y are calculated from (2-5). Problem.Statement Given a class Of large. nonlinear. and dynamic models. the prob- lem addressed in this thesis is the construction of an optimisation component that will efficiently assist in Obtaining good parameter estimates or controls. Good parameter estimates are those consistent with all known characteristics about the system behavior while good controls are those that realise the desired behavior Of the system. The Optimisation component is a set of algorithms that search for good estimates or controls by solving one or more optimisation problems in an interactive mode. Efficiency of the component is measured by fOur criteria: (1) the costs to the user. (2) the convenience for the user. (3) the compatibility of the component with various computers. and (4) the accuracy of the problem solution. In a specific applica- tion an overall measure of efficiency can be fOrmulated as e : E {f(costs. convenience. compatibility. accuracy} 2-11 20 where f is an explicit measure of efficiency given cost. convenience. compatibility. and accuracy of solution measures. and E is the expected value Operator taken over the class of models of interest. Though an explicit expression for e is not developed or used in this thesis. in specific applications different components can be evaluated by their relative values of e. Since large simulations are expensive to run on a computer it is important. in using the component. to reduce costs when possible with- out sacrificing convenience Of use. computer compatibility. or accuracy of solution. The costs to the user accrue from fOur principle sources: (1) the number Of simulation runs necessary for the optimisations. (2) the amount of central processor (CPU) time and (3) core storage used fer the calculations. and (4) the accuracy of the optimisation solution. or the error between the component's solution and the true solution Of the Optimisation problem. There are many tradeoffs be- tween these sources Of costs. Fer example. an increase in the accura- cy of the Optimisation solution can usually be Obtained at the ex- pense of CPU time. Considerable savings in computer costs can be re- alised from.a reduction in either the number of simulation runs. amount of CPU time. or core storage used. or an increase in the accu- racy Of the optimisation solution. In view Of the above. the measure of efficiency can be made more specific: e : E {f(g(n. t. c. a). c1. 02. 03)} 2-12 where n=the number of simulation runs e=core storage used t=CPU time used 21 a=accuracy of optimisation solution C1=a convenience measure to computer compatibility measure and C3-an accuracy Of problem solution measure The importance of convenience of use as a criterion for measuring efficiency cannot be overemphasised. Convenience is manifested in the capability of the component to easily interact with the user. This interaction is important from at least fOur aspects: (1) different variable sets may need to be examined. (2) several sub-problems re- sulting from multiple performance indices may need to be solved. (3) infbrmation obtained about the simulation from an optimization may indicate further investigation of specific model behavior and. (4) parameters of the optimization methods in the component may need to be changed to increase the efficiency of the component. These aspects can be examined in an Off-line or batch mode of Operation on the computer. However. an interactive capability of the component will reduce the direct costs of Operation. provide the user with a potentially rich experience in understanding the behavior of the simulation. and permit an examination of intuitive judgements or "hunches" about the solution Of the problem to be made before being fOrgotten. It is also important that an optimisation component. a computer program that could be used in a variety of situations. be compatible with a large number of computers. This means that the component should be capable of execution on other computers with virtually no changes in programming. The most important aspect of compatibility is the choice of programming language to be used. FORTRAN was chosen 22 because of its extensive use in science and engineering. its capabil- ity fOr execution on relatively small computers. and its versatility. Other factors. such as the amount of core storage used and special features of the programming language must be taken into account in the decision Of how much compatibility the component should have. Large savings in costs. resulting from reduced real world system costs. can sometimes be realised by an improvement in the accuracy of the problem solution. Improved parameter estimates result in accurate models. and improved controls result in efficient system Eigoiition. Thus. better ways in which to design. develOp. or control the real world system.can be realised by experimenting with the model if the accuracy of problem solution can be improved. The development of an Optimisation component that "best" satisfies these fOur criteria fer efficiency. one that optimises e fer a partic- ular class or set of models. is the problem examined here. Clearly it is impossible in a practical sense to perfOrm an optimisation of e for an infinite class of‘models. HOwever. in the development of a component fer a particular class it is important to examine potential candidates fbr a component by evaluating e fbr a number of representa- tive models. Lacking an explicit fOrmulation fbr e. as occurred in this thesis. a subjective judgment on the suitability of a component was made on infbrmation obtained from (1) the literature. (2) potential users of the component. (3) possible classes of models on which the component will be used. and (4) experiments with different components on rep- resentative models. From.this information three factors Of efficiency were heavily weighed in this thesis to determine component efficiency: 23 (1) the number of simulation runs necessary fer convergence. n. (2) theme“ of use of the component. and (3) the accuracy of the problem solution. Review Of_2revious werk In the fields of mathematics and engineering considerable research has been conducted in the area of function Optimisation. As a result. numerous search techniques have been develOped to optimise functions when only function evaluation is possible. It is impossible to de- scribe all Of these methods or even a fraction in any detail. the lit- erature being extensive in this area.2'3'm'3o However. a brief sur- vey Of the more pOpular and representative methods will be examined here. A detailed description Of the Complex method and Powell's method is included because these are two methods used in the component de- scribed in this thesis. Also. fOur previous attempts at constructing an efficient optimisation component will be discussed. Optimisation Methods There are two classes of optimisation methods that have been used in Optimisation problems. These are (1) direct methods in which only function values at points in the variable space are used in the search and (2) gradient methods in which the gradient. or an approximation to it. is used. Each of the methods in either class have certain advantages and disadvantages that will be discussed. Many of these methods employ linear or one dimensional searches which are discussed in the literature.2'3'1u'3o Direct Methods There are two types of searches that comprise this category. first and second order methods. The order corresponds to the information 24 about the function that is implicitly used in the method. First order methods use information generated about favorable directions of search while second order methods use additional infbrmation about how the favorable directions are changing as the search proceeds. The dividing line between what constitutes a first or second order search is vague with many methods being classified as belonging to either class. Rep- resentative methods of the first order include the univariant.3 simplex.3 complex.3'lu and HookeeJeeves.3 Powell's 2.3.5,14,2u.25 and Rosenbrock'damethods are examples of second order searches. Univariant Search3 The univariant search consists of one dimensional searches along the coordinate axis. In this method. from an initial base point. searches are made along the coordinate directions using the Optimum along one direction as a base point fer the search along the next direction. After the n coordinate directions are consecutively searched. in an n dimensional problem. this method resumes by search- ing over the directions again. The algorithm terminates when no further significant improvement in the Objective function is possible. There are several disadvantages of this method: (1) it becomes increasingly difficult fer the method to converge as a local Optimum is approached. (2) the method may terminate along a ridge or valley rather than converge to a local optimum. and (3) convergence is slow compared to other methods when the function has interaction in the variables. Although the univariant method is generally a poor choice for use in solving an Optimisation problem because of the disadvantages men- tioned. it is described here because it is a part of some of the 25 components described later. Simplexnethod3 The simplex method is easy to program and places a small demand on core storage. The method evaluates the function to be Optimised at the vertices of a regular simplex (equisided geometrical figure of n+1 points in an n dimensional space). The vertex with the highest func- tion value (in a minimisation problem) is replaced by its reflection about the center of gravity of the old simplex and a new simplex formed with the new vertex. The function is then evaluated at the new vertex and the process continues until no improvement is possible. Then either the search halts or a smaller simplex. constructed from.the final one. continues the search. If the search halts. the vertex with the lowest function value is taken to be a local minimum. Two addi- tional simple rules ensuring convergence of the method in most situa- tions are used. In this method selection of a new point. or vertex of the simplex. requires very little computer time while only the vertices and their function values need be stored in the computer. In addition. since the search uses only function value comparison and not relative func- tion value changes to locate improved points. the simplex method works well when the function evaluation is corrupted with random errors. There are several disadvantages of this method: (1) no term of acceleration along favorable directions of search. (2) poor ridge and valley fellowing characteristics. and (3) no reliable way of dealing with constraints. 26 Complex3’1a The complex method is a modified version of the simplex that in- cludes an acceleration feature. NO longer is the n dimensional figure constrained to be equisided. but it is allowed to contract or expand along favorable or unfavorable directions of search. Like the simplex method this method also uses little computer time and storage. Since the complex. n91 points in an n dimensional space. is not necessarily regular it can effectively cope with constraints on variables. The rules fer this method are: 1. An initial feasible complex is termed of n+1 points chosen randomly around an initial guess to the Optimuma 2. If the constraints are violated by a vertex of the complex during the search that vertex is moved a small distance inside the feasible region. 3. The objective function is evaluated at the vertices yielding a point field with the highest function value. This point is replaced by a new one. zhew’ given by g . - 2- anew ‘ (15c 501d) + 5c 13 where 1 2 ,___. x - 5c n i+<221d 1 * 2 1“ is the centroid of the remaining vertices of the complex.‘1_c1 is a vertex Of the complex. and a is an acceleration factor usually chosen to be 1.3. 4. If a point repeats in having the highest value on two consecu- tive trials it is moved one half of the distance to the cen- troid Of the remaining points. 5. The search continues until a predetermined maximum number of 27 iterations is reached. a convergence criterion is met. or it is externally halted. The complex method is a highly effective first-order method on a variety Of functions at large distances from a local. optimum.5 Like the simplex method. it performs satisfactorily with random errors in the function evaluations. However the complex method is not as effi- cient as the direct second-order methods near an optimum. Hooke-Jeeves Search3 This method. while not suitable for a broad class Of problems. has certain characteristics which are incorporated in the component de- scribed later. This search is effective with functions having rela- tively straight ridges or valleys. In the search. local explorations of the variable space are coupled with acceleration along favorable directions. This popular search is well documented in the literature. Powell's Methodz'a's'w'zu'zs Powell's method is an example of a second-order method. It con- verges well in the vicinity Of a local Optimum. The search is based on the fact that the minimum of a quadratic function in n variables. f(x1. x2. .... x") = f(x_) = xtgygtxeb where Q is symetric positive definite. can be found by searching once along each of the directions 31. 32. .... 2n if 3:933 =-. O. for 14:). These directions are said to be Q- conjugate or conjugate with respect to Q. Thus only n linear searches are required to find the global minimum of a quadratic func- tion. With a nonquadratic function f(_x_) in n variables more linear searches will be required to find a good approximation to .‘ local minimum. In the search an approximation to the Hessian. or matrix of 321' \ 3 x1 3 x3 I second partials. ( corresponds to the matrix Q with search 28 directions constructed so as to eventually be conjugate with respect to the Hessian at the minimum. The method of search begins with an initial set of directions. usually chosen to be along the coordinate axis. and an initial point £0. Then linear searches are made consecutively along the directions using the Optimum of one search as the starting point of the next search. This results in a final point. i. being the Optimum along the last search direction. The search then forms an expanded point 5. = 2331350 and tests whether‘ng.) _>_ “50). If so. a test is then made to determine if a valley is present in the direction Of £115 . This test is: - 2 (Po-ZF‘eFJ (Po-1"- A )3. 9:921") with A = IFi-F1-1| 2-15 F1 is the function value of the point with the greatest function im- provement between linear searches. If this test is satisfied or f(J_t°) f_ f(_x_o). then the direction 5.150 crosses a valley and the old search directions are retained with 5‘ the new starting point. If the test is not satisfied. then the direction 5"5-0 is introduced to replace one of the original direc- tions and a search made along f-xo locates a minimum point to begin the next iteration of linear searches. This search is based on the fact that the direction inf-50 f(_x_) = 5"? ':T . is Q conjugate to a space (or all the directions .if in the space) in which 50 is the minimum. This is proved in the fol-- lowing theorem.2 Theorem: If fix) a: 162/2 9 at! e b with cht > O and f(x ) a: min f(3_t_) . “5.) a min f(§) . where MCN. '0 36:: gm then (Er-mfg a: O for all 5814. 29 Proof: since Zf(_x_o)°;_t_ a 0. 55M and zfQED-x = O. 3N then 2f(z')-zf(zo)’§ = 0 for 56M or “5.190%! = O. and (i-xofigx = O, xcM sincegcgt' In the search 14 corresponds to the space in which go. the starting point fer an iteration Of the search. is a minimum. At the k'th iteration of the search with f quadratic. M is the space spanned by the k directions of search that have been introduced since the search started. N is the space spanned by the k directions and g-x . Thus. .0 by the theorem. the new direction 5550 is Q conjugate with respect to the k directions and there are ke1 Q- conjugate directions to begin the kei iteration of the method. The search continues until satisfactory convergence is Obtained eg. II xi-x1+1|li€' where I] ”is themaximumnorm. This method has excellent convergence properties if the function to be optimised can be satisfactorily approximated with a quadratic function at the local minimum. This is very desirable from a practi- cal viewpoint as many functions can be adequately approximated by a quadratic in local regions of subsets of the variable space. The literature presents a strong case for the use of this method with a variety of functions. Rosenbrock's Method2'3 The general idea of Rosenbrock's search is to align the directions of search along directions that produce the greatest changes in the function. As in the univariant search the variables are varied one 30 at a time along the most recent set Of directions. New sets of search directions are formed from the introduction of one new direction at each iteration of the method and constructing the others so that they are all mutually orthogonal (using the Gram-Schmidt process). Rosenbrock's method has found wide pOpularity. however it is slow in starting and has no satisfactory convergence criterion. Gradient Methods (Modified Davidon's Method)3 The basis for the method is that if one has a quadratic function to minimise. f(1_c) : gtg/Z e gt; 9 g with g symetric positive defi- nite. then the minimum of f(x) occurs at a; = 5-9-15. if 9.1 is avail- able. If Hg) is not quadratic 9 corresponds to the Hessian matrix. Davidon' s method constructs from an initial approximation to the in- verse of the Hessian at a point. a sequence of matrices that converge. under suitable conditions. to the Hessian at the local Optimum. The modified version uses approximations to the gradient vector 3 .. 1%:Wy with t5 small and _e_‘1 a unit vector in the i'th coordinate direction. This method has extremely good convergence properties near an Optimum. However. random errors in the evaluation of the function f(;_:) considerably lowers the convergence rate as compared to other methods utilising second order infbrmation. In addition. the number of simulation runs necessary to calculate the gradient tends to make this and other gradient methods less efficient than direct methods when the gradient is not available in an analytic form. Optimisation Components An optimisation component is a set of methods and techniques that retains the advantages and eliminates the didadvantages of individual 31 methods by ferming an efficient integrated approach to an Optimisation problem. Several components have been developed and used in optimi- sation problems. The feur that are described here exemplify the ap- _proach taken and provide motivation for the component developed in this thesis. Univariant-Polynomial Appgoacha'u This approach involves a two-stage component consisting of a uni- variant search and a polynomial approximation. usually quadratic. on the function to be Optimised. In this component the univariant search is executed until convergence is obtained or the method halts. A polynomial approximation is used to determine a new direction to search if a hyperplane was fitted to the function. or where the Opti- mum.is located if a higher-order polynomial was fitted. If desired this entire process is repeated until a specific convergence criterion is satisfied. While this component overcomes many of the disadvantages of either method used separately. other problems remain. The component relies on ferming efficient experimental designs to ease computational com- plexity and reduce the expected variance of the fitted polynomial to the function. Thus. the component depends. at various stages. on pre- planned experiments which tends to reduce the infermation Obtained about the optimum in comparison to that Obtained in a strictly se- quential method. This emphasises another problem with this approach. The univariant search and polynomial approximations are not compatible techniques in that information obtained in the execution of one cannot be used to reduce the amount of computation in the other. Though this method will converge in most applications. it is 32 slower to converge than other methods comparable to it in complexity. Sigplex-Polynomial Approach3 This two stage component is similar to the preceeding one except that the univariant search is replaced by the simplex algorithm. This was done in order to retain the advantage of a sequential search while allowing fer the construction of the polynomial approximation with no additional function evaluations. This results from the simplex method naturally ferming an efficient experimental design during a search. Like the simplex method. this component does not have the ability to accelerate the search along favorable directions. In addition. the component is heavily dependent On polynomial approximations and thus ‘will not be efficient if the surface cannot be adequately approximated by a low order polynomial. Sim Optimisation12 Sim Optimisation is a three stage optimisation component consisting Of (1) a decentralised gradient approach (DGA). (2) a linear response surface approximation (LBS). and (3) a quadratic response surface (0R3) approximation. ‘Which stage is being executed depends on how fast the function is changing. The decentralised gradient approach is a decomposition procedure for creating from one difficult to solve large problem. many smaller sub-problems which are easier to solve. The procedure uses the solu- tion of the sub-problems and infermation of how they are related to construct another set of sub-problems to be solved. The new set of sub-problems have a solution which is generally closer to the solu- tion of the original problem. The procedure is repeated until no further improvement in the solution is possible. 33 The LRS and QRS stages consist of approximations of the objective function by a linear or quadratic polynomial. depending on whether an improved direction of search or an improved estimate of the local Opti- mum is desired. These stages are similar to the polynomial approxima- tions in the two components previously described and they have the same problems. Although the BOA stage is capable of quickly locating regions in the variable space which are close to a local Optimum. unfortunate- 1Y' the component cannot be applied in a wide variety Of interesting problems. In addition there is little capability for interaction with the component. 292.2229 CADSIS. an acronym for computer aided design of systems. is a two- stage Optimisation component that is executed on a computer in an interactive mode. The user of the component can change the Objective function under study. constraint levels. or search variable sets during the programs execution. In addition. specific behavior Of the model can be examined or optimisations perfOrmed on command of the user. The component incorporates two search methods. Hooks-Jeeves and Jacobson-Oksmar? (a special case of which is Davidon's method). The Hooks-Jeeves pattern search is used to start the search and when in the vicinity of the Optimum (the objective function changing slowly) the Jacobson-Oksman method is used fer further improvement. In this component either a single optimization problem or a sequence of problems can be solved. ‘While the component was original- ly constructed to aid in the design of systems. it can also be used to 34 help solve parameter estimation and policy decision problems. Component Requirements The problem statement and the literature review indicate certain requirements that an efficient component should have. These require- ments follow from the consideration of the four factors measuring efficiency (1) costs. (2) convenience. (3) compatibility. and (4) accuracy. and the disadvantages of individual Optimisation methods discussed in the previous section. There are nine requirements: 1. the capability Of effectively cOping with multi-optima 2. the capability of dealing with noise in the objective function evaluation 3. in a parameter estimation problem. the capability Of dealing with noise in the real world data. possible with non-zero mean and correlation 4. compatibility between optimization methods used in the component 5. a capability Of approximating the objective function globally or in particular regions with a polynomial 6. a search algorithm that is effective both far from and in the vicinity of a local Optimum 7. the capability of examining specific model behavior 8. the capability of sensitivity testing on the Objective func- tion 9. interactive capability to change the objective function. constraint levels. search variable sets. and optimization method parameters The next chapter contains a detailed description of the optimiza- tion component developed in this thesis. Its construction and design will be discussed in relation to the component requirements and the problem statement examined in this chapter. CHAPTER III DESCRIPTION OF THE OPTIMIZATION COMPONENT This chapter presents both a general and a detailed description of the optimisation component developed for this thesis. The general description consists of a non-mathematical discussion of the overall structure of the component and its relationship to the other elements of the problem such as the simulation. the real world. and the user. The detailed description treats the internal structure and organisa- tion of the component. Each of the subroutines contained in the com- ponent is described in this chapter. Furthermore. subroutines repre- senting significant thesis contributions are discussed in depth. A rationale or defense of the component construction in light of the problem statement and component requirements presented in Chapter II is given. In addition. a summary of the contributions or new developments of the component to research in solving parameter esti- mation or policy decision problems with numerical optimization methods concludes the chapter. General Description The behavior Of a model can Often be studied or improved by print- ing or plotting real world or model time series and by solving a se- quence of user specified numerical optimization problems as discussed in Chapter I. With the aid of simulation. the Optimisation component presented in this thesis can be used to examine model behavior or 35 36 solve complex model generated problems. The user of the component. either the model builder or policy analyst. can be assisted to effi- ciently solve parameter estimation or policy decision problems. To accomplish this the component is constructed in two sections (1) a pattern recognition sub-component and (2) an Optimisation sub-compo- nent. The purpose of the pattern recognition sub-component. as its name implies. is to determine if the model exhibits reasonable behaviorial characteristics and also to select regions of the input variable space fOr future investigation in the optimisation section of the component. Thus. patterns Of behavior of the model are used to aid in solving the parameter estimation and policy decision problems. This sub—component is designed to allow the examination of the behavior of a simulation in a batch or an interactive mode of execution on a computer. Because accuracy of the problem solution can be improved in a parameter esti- mation problem if filtering techniques are employed,this sub-component incorporates data smoothing algorithms. In addition. it is important to be able to study the model outputs for different sets Of values of the input variables. Therefore. data smoothing of real world time series and printing and/or plotting of model time series are two of the major capabilities incorporated in this section of the component. The optimisation sub-component is designed to be executed only in an interactive mode with the user and with several possible levels of interaction. The different levels Of interaction are included to al- low the user to select an appropriate fermat for the study of the model. This section of the component also contains features useful in solving many of the problems associated with nOnlinear static 37 optimisations of complex functions. The Objective function. search variable sets. Optimisation method parameters. and constraints on the variables can all be changed during the execution of the component. In addition to being able to direct various Optimizations to be per- formed. the user can also study the sensitivity Of the Objective func- tion to changes in the input variable values. The purpose of this sub-component is to solve numerical optimization problems that will lead to good parameter estimates or policy decisions. The flow chart in Figure III-1 illustrates the data flows between the simulation model. the Objective function. the real world. the user and the optimisation component. It is important to realise that only one sub-component is in execution at a given time. In general. one would investigate model behavior characteristics first with the pattenu recognition sub-component to Obtain a problem solution. However. in many situations an iterative process is necessary to Obtain good prob- lem solutions where the pattern recognition and Optimisation sections of the component are each called into execution a number Of times. As a result of the availability of the CDC 6500 computer fer the research conducted fer this thesis. certain features of the use and Operation Of the component are particular to this computer. However with relatively small programming changes. (see Appendix) the compo- nent should be executable on a wide variety of computers. The compo- nent can be stored in the computer on a magnetic tape or disc. It is only necessary to have the simulation to calculate the model outputs. and two user supplied subroutines to specify the ferm of the objective function. stored in the computer in order fer the component to execute properly. Once this is done the execution of the entire component human interaction controlled ttern nent variables - P‘ °°'P° ___ simulation mocel recognition £0201.) Mdel on") 'E section outputs unconttolled variables pa ameter estimation ._._:_n. : a- real - objective — —- < — .) -— "0:15 section _______ function — sys em 3531‘) i - - noise interactive component variables outputs human interaction 2,1s the parameter set that specifies the Objective function J g|is the set of model parameters being studied g'is the model control variables being studied T is the length of the time horizon of the model [b.f] Figure III-1 STRUCTURAL CONNECTIONS TO COMPONENT 39 can be co-ordinated from.a teletype terminal with any large plots or printout from the pattern recognition section being disposed to a batch printer. The design of the component is general enough so that no programming changes need to be made to change the investigation from one simulation to another. An example of the operation of the component from a teletype and a guide to its use is provided in the Appendix. The next section of this chapter will provide a detailed descrip- tion of the component's overall organisation and the various sub- routines included in it. Detailed Description The component is described in two parts (1) the pattern recogni- tion sub-component and (2) the Optimisation sub-component. Both sub-components represent complete computer programs capable of inde- pendent execution. The pattern recognitiOn section consists of a main program with three subroutines while the optimisation section contains a main program and fifteen subroutines. Together they occupy 12.000 decimal words of core storage in the computer during execution. A detailed description and flow chart of the structure and logic of each sub-component and major subroutine is presented. Pattern Recognition Sub-Component The pattern recognition sub-component is constructed to give maximum flexibility to the user of the component in examining model behavior. This part of the component contains a main program. MAINPR. and three subroutines. SUBPR. SMOOTH. and PLOT. The main program inputs key variables that affect the sise of arrays needed in the re- mainder of the program. In addition. a method to control variable no dimensioning Of arrays is included in MAINPR so that core storage is minimised during the execution of the program. SUBPR. the heart Of the sub-component. contains the majority of input-output functions. data transfers. interactive decisions. and logic execution in the program. SMOOTH is a subroutine designed to remove random fluctuations from the real world time series with a first-or second-order fixed memory polynomial filter. Finally. the subroutine PLOT performs the function of plotting. with or without automatic scaling. or printing either model outputs or real world time series as functions of time. The flow chart of subroutine calls and interactive decisions in this sub-component is contained in Figure III-2. This flow chart illustrates the overall structure and logic flow of the sub-component. The subroutine SIM in the flow chart is the simulation model which must be in a specific ferm in order to correctly interact with the component. This fOrm is typical of a simulation of a dynamic model and is discussed in depth later in this chapter. The capabilities available to the user which can be obtained by fixing the values of certain variables in the program include: (1) Smoothing of real world time series with a (a) first or (b) second order polynomial filter. (2) (a) Printing or (b) plotting all or a subset of the model outputs for any specified set of values Of the controlled variables. This can be done for different time intervals as specified by the user. (3) Execution Of the simulation in the normal mode (see section on user supplied subroutines). 41 ? ®éfl_0 PRINT rm sauna: 30393 A so. mm 3157134 A VARIAsLns J, not PARAMETER ( : ) Tl, ss'rnmmn' o c on ‘ ° >® PLOT (R OPTIMAL @ém mm oomnor. MODEL PROBLEM cursors? Jinn. INPUT - REAL WORLD no CHANGE rm); 3mm; DEPAULT VARIABLE \I’ _ nuns: IS snoommc OF TIME .32.... spams canes DESIRED 03mm VARIABLE ‘ VALUES V [—SMOOTH B A Figure III-2 FIN CHART OF THE PATTERN RECOGNITION SUB-mom INPUT VARIABLE NAMES TO BE PLOTTED OR PRINTED l SIM PIDT no saunas snnm 3mm SIM A CONTINUE SIMEX STUDY? Figure III-2 (cont'd) CONTINUE PATTERN RECOGNITION STUDY 1 N0 V WRITE INI'ORMATION NR OPTIMIZATION SUB-COMPONENT INPUT FILE RITE TAPE 98 'END 43 (4) writing on a permanent file data to be transferred to the Optimisation sub-component. Proggam.MAINPR Program MAINPR inputs. in either a batch or an interactive mode. the variable values that fix array sises necessary for proper compo- nent execution. All arrays in the program are formed as parts Of one large array stored in BLANK COMMON. In this program. space is allo- cated as needed for a particular problem or simulation model. Fixing the exact amount of space required for the program results in time and cost savings to the user as recompilation of the program is not neces- sary to properly interact with different simulations. In other words.' with no programming changes the capability Of variable dimensioning during execution allows the use of the component with different simu- lations. Figure III-3 is a flow chart of MAINPR. Subroutine SHEER This subroutine handles all the necessary input of simulation model parameters. decisions based on interactive variables. and logic execution of the sub-component. Calls to the subroutines that smooth the real world time series. print and/or plot time series. and calcu- late model outputs are all made from SUBPR. The flow chart of the pattern recognition sub-component (Figure III-2) is essentially of the structure and logic of this subroutine and therefOre a detailed flow chart Of SUBPR is not included. As indicated by the flow chart of the sub-component. there are many data manipulation and plot/print options available to the user. In the sub-component it is possible to compare in either a graphical or tab- ular manner real world data. model outputs. and smoothed time series. 44 ( START MAINPR > DIMENSION AN ARRAY TO LENGTH 1 IN BLANK OOMPDN i READ VARIABL$ AFFECTING SIZE OF ARRAYS IN SUB-COMPONENT CALCULATE STGZAGE LOCAT IONS FOR ARRAYS IN BLANK COMMON EXTEND (”MMON TO SIZE CALL SUBPR ASSIGNING ARRAYS TO PROPER STORAGE LOCATIONS FOR THE EXECUTION OF THE PROGRAM END MAINPR Figure III-3 now CHART 0F MAINPR “5 Thus. decisions can be made as to whether or not to use the actual or smoothed time series fer further investigation with the optimisation sub-component. In either a parameter estimation or policy decision problem the graphical and tabular output can be used to examine the' behavior Of the model to determine whether the model exhibits reason- able behavior and also to isolate feasible regions of variable values for further study; Included in SUBPR is the Option of writing on a permanent file in the computer system variables such as input and output variable names. and real world and smoothed time series. This file can then be stored for later use with the Optimisation sub-component. thus minimising the input of duplicate information to the different sections. Subroutine SMOOTH Preliminary tests of the use of numerical optimisation methods to solve parameter estimation problems were attempted using a model of the Nigerian cattle industron. The real world time series used in the problem were taken to be model outputs generated from a base parameter set 3 and corrupted with an additive noise component. It was found in these tests that smoothing or filtering the real world time series (artificially generated) allowed for reduced errors. 2. between 2_and the parameter estimates computed by minimising a sum of squares error criterion. In addition. it was also discovered that if the smoothing polynomial was selected to match the general curvature of the real world time series then the errors 2 could be reduced further. Typically. smoothing with the appropriate polynomial reduced the mean of the errors 2_ by an average Of 20-30% in this test prob- 101‘s 46‘ Smoothing of the real world time series is accomplished in the sub- component with the use of a first or second order moving polynomial filterzo. The filter uses Legendre orthogonal polynomials to reduce the amount of computation necessary to perfOrm the smoothing. Defining 1 (391) c = 3-1 3 (23+1)L(3) where L01 2 the number of data points used for the smoothing and 1(3) = I(I-1) (I-Z) (I-3)......(I-jel) ‘with I an integer. Let the L91 data points to be smoothed by the polynomial filter be Yn--I.' yn-Lu' ""331 and define a polynomial of order 3 w()- 1 :(1)k(3) (3k) x(1c) 2 a“°ak,o' k kw 3' then it can easily be shown that n n £6 £0 “3m W1“) = 611 3‘3 where 613 is the Kronecker delta. If m.is the order of the polynomial filter. then the smoothing poly- nomial is defined to be (p‘(x))n. the polynomial of degree m that minimizes a sum of squared errors criterion. en. where L 2 on = :20 (Yn_L+x-(p(X))n) 3-“ with >n = :30 (bpn w (x) 3-5 To find (13‘ (30)n that minimizes en one sets den = 0 for j = O. 1. .... m d. EXIT FROM PROGRAM? CHANGE DEFAULT VALUES 7 INPUT DEFAULT VARIABLE VALUES RETURN 9'? DATA! 1 \LNO RETURN C? DATA2 1 N0 Figure III-5 (cont'd) RETURN DATA3? RETURN SENS TESTING r RETURN “TA“ 53 so: or 30TH FUNC- Ee‘ EVALUATES 2. secsw- 31mm- THE RRERARES ARRAIS , RREmRMs Eé—J OBJECTIVE EUR SEQUENTIAL 9 SEQUENTIAL FUNCTION RD‘SRESSION REGRESSION FOR J.- m J Rat POI-WON“!- 1. COEFFICIENTS \ TRANSFE- ASSIGNS n. VARIABLI .____9 sm .___.9 UPDATES not FOR EVERY m STEP SIMULATION CALL SENS FUNC TRANSFE SIM T [5 [:33 En; m Figure III-6 MAILED FLOW CRARTS OF THE SUROUTINB ml. m. SE15. AND SPEED r 8313 PUNC l , I i I an: [UPDATE [Rat i _ [ UPDATE det Figure III-6 (cont'd) 55 The following descriptions of the subroutine in the program should be related to these flow charts to obtain a clear understanding of the operational aspects of this sub-component. Program.MAINOPT The main program in the optimization section is similar to the main program. MAINPR. in the pattern recognition section of the compo- nent. Dimensioning of arrays whose sizes depend on the simulation being studied is performed in this part of the program. Once these arrays are dimensioned their sizes remain fixed during the entire study of the model. To complete execution of the program. control is next transferred to the subroutine SUBOPT. Subroutine SUBOPT This subroutine is similar to SUBPR in the pattern recognition section of the component in that it contains the basic logic and in- teractive capability of the sub-component. Arrays which are problem dependent. those whose sizes depend on the number of search variables. are properly dimensioned in this subroutine. This allows the reduc- tion of core space used to the minimum size necessary for a given problem at a given time. Much of the important output of the component is printed from this subroutine. This includes local and global optimum.information. sen- sitivity analysis results. and running totals of program execution costs. In addition. the subroutine calls to input or change data for program: mution or problem construction. to perform numerical optimi- zations. and to test the sensitivity of the objective function to changes in the input variables are all made from SUBOPT. The flow chart of the optimization sub-component in Figure III-5 56 is essentially that of the logic flow and structure of SUBOPT. Therefore. a detailed flow chart of this subroutine is not included here. Subroutines DATA1._DATA2. DATA}. DATAQ. and VAR This set Of five subroutines fOrm a data input hierarchy within the sub-component. Each subroutine reads from a teletype terminal different types of data necessary fbr the execution Of the program according to the user's needs. Subroutine DNTAI inputs to the program two variable sets (1) the model variables and (2) the interactive variables. The model varia- bles include any input or output variables of the simulation being studied that are necessary for calculating the objective function. In addition. the set Of all possible search variables in which the user is interested is considered to be a part of the model variable set. Included in the interactive variable set are those variables specify- ing the fOrm. type. and limits Of the Optimization searches used in the component. ‘With these variables an Optimization can be structured to match the user's needs fbr the amount and type of output. inter- action. and numerical methods employed in the investigation. This structuring is perfOrmed by setting the values of the interactive variables to certain constants. The instruction array. IS. is a large subset of the set of interactive variables and determines what Options will be included in the execution of the sub-component. The array elements and correSponding Options are displayed in Table III-1. The subroutine DATAZ inputs the variable set consisting Of the parameters of the numerical Optimization methods. These variables directly affect the performance Of the Couplex. Powell's and Powell's Table III-i Instruction Array nsments and Corresponding options ELEMENTS 15(1) 15(2) 15(3) 13(5) 13(7) IS(8) 13(9) IS(iO) 13(12) 13(15) 15(16) 13(18) OPTION 0 for a parameter estimation problem 1 for a optimal control problem 0 to print coefficients of approximating polynomial to the Objective function 1 not to print coefficients maximum number of local searches to be conducted fer the global search ' O for no speed up of Powell's methods along a sharp ridge in the Objective function 1 for execution Of the speed up algorithm 0 if sensitivity testing of the Objective function in ' the entire input variable set 1 if only sensitivity testing on subsets of the input variable set is desired 0 for no interaction (selecting initial points for local searches) in global searches fer optional interaction in global searches fbr no polynomial approximation to the .ObJective function to be calculated fbr a local polynomial approximation fbr a global polynomial approximation ONO-.OH for regular switching between the Complex and Powell‘s methods 1 fer switching with construction of approximate conjugate initial search directions fbr Powell's method 0 fer swiching interaction between the Complex and Powell's methods after every iteration of the Complex method en fOr switching interaction after n iterations of the Complex method -n fbr automatic switching after n iterations of the Complex method en fbr an output interval of n iterations in the Complex method 0 fbr no interaction in Powell's method 1 fOr interaction (to exit) after every iteration 2 for interaction after every linear search 3 fer interaction after every function evaluation 0 fer a least squares error criterion in a parameter estimation problem 1 for a normalized least squares error criterion in a parameter estimation problem. 58 speed-up methods. Examples of variables in this set are the reflec- tion constant in the Complex method. and the step size scaling parame- ter in Powell's method (see later subheadings on BOX and BOTH sub- routines). The parameter set. 2, that fixes the form Of the Objective func- tion J is input in DNTAB. As discussed previously. it is important to be able to change the form of the Objective function being studied when a single function cannot be constructed to measure how'well a system perfOrms. The subroutine DATA“ inputs to the program the specific variable search set Over which the optimization is to occur. Any constraints on the variables are also read in this subroutine. The search varia- ble set must be a subset of the input model variable set read in DATAL The subroutine VAR. which is executed immediately fbllowing DATA“. perfOrms the function of checking that the variable search set is actually a subset of the input variable set. Also VAR allows fOr changes in the search set to be made if errors are detected. As can be seen from the flow chart Of the optimization section. a return can be made. at the user's option. to any of the DATA subrou- tines during the execution Of the program. This allows the variable search sets. the constraint levels. the Objective function. and the type of numerical search to be changed while the program is executing. To accomplish this. recompilation of the component is not necessary. ‘With the variable dimensioning capability of SUBOPT and the capability of changing the Optimization problem under investigation. a minimal demand on core storage is made while retaining the Option Of studying a diversity of problems Of interest to the user. This subroutine 59 hierarchy provides for a flexible and convenient. but powerful. format for changing variables values in the component. Subroutine SENS Sensitivity testing on the objective function is performed in this subroutine. If the objective function is dependent on the input variable set {3. g. '1'} then the sensitivity of J*[9_. g. 'r] with re- spect to a variable x({g. u. r} is defined to be sx where AJ’ _ chaeinfduetoachangeinx_ J.l °x-"—-L£cmg.m ’33:: 3'8 x with AJ’the change in J'resulting from the change in Ax in the variable x. Sensitivity tests on the objective function identify those varia- bles that most strongly affect changes in the performance of the model. As a result. these tests are important in the solution of parameter estimation and policy decision problems for at least four reasons. First. the relative sensitivities of the variables may indicate search variable subsets that could allow for rapid and large improvements in the objective function. Second. the input variable sensitivities may show where a good initial point for a search is located. Third. by evaluating the sensitivities at different points near a base set of values some interactions in the variables may be isolated. This would allow a reduction in the size of the search variable sets necessary to find an optimum of the objective function. As a result. a corre- sponding reduction would occur in the number of simulation runs needed for the optimization. Last. a sensitivity analysis at the optimum in a parameter estimation problem can assist in determining whether the model is a valid representation of the real world18. 60 In the subroutine SENS the sensitivity of the objective function can be calculated in all the input variables (requiring n91 simulation runs with n input variables) or to subsets of the entire variable set. The calculation of sensitivities of important subsets can eliminate unnecessary simulation runs. The desired mode of execution can be specified by an interactive variable input in DATA1. Subroutines BOX. BOTMl and TRANS The subroutines BOX and BOTH are the FORTRAN programmed versions of the Complex method and Powell's method described in detail in Chapter II. These subroutines are modified versions of programs avail- able from a commercial optimization packageIa. Changes made in the program consist of additional printing and interaction options. inde- pendent selection of the initial set of search directions for Powell's method. and automatic generation of initial complexes for the Complex method. The subroutine TRANS determines when a switch is to occur between the Complex method and Powell's method. Various options. including both automatic and interactive switching. are available to the user through the input of an interactive variable. The general question of when to most efficiently switch from a coarse search to a fine search is an area needing further investigation. There are. however. large potential savings in computer costs that could result from the use of an appropriate decision algorithm. In the Complex method it is only possible to interact with a search after a predetermined iteration interval. while in Powell's method there are three points of potential interaction (1) after every function evaluation. (2) after every linear search. or 61 (3) after every iteration. The type of interaction desired in Powell's method and the interaction interval for the Complex method can be chosen through the input of an interaction variable in the DATA subroutines. The Complex:method and Powell's method were selected for use in the component for several reasons. First. the literature review and ini- tial experimentation5 with the various methods indicated that the Complex.method was an effective search algorithm to quickly locate a region of the variable space containing a local optimum. In addition. Powell's method produced rapid convergence in regions of the variable space in which a local optimum existed. Therefore. by combining these two searches into one two level optimization algorithm some of the disadvantages of the separate methods were eliminated and the advan- tages retained3 . With the two level strategy good convergence could be obtained in broad areas of the variable space. Secondly. it was possible with these two methods to achieve com- patibility with the use of conjugate directions calculated from a quadratic polynomial approximation to the objective function. In this way information generated about the behavior of the objective function in the Complex method could be incorporated in the initialization of the search directions for Powell's method. This generally resulted in a rapid initial convergence rate for Powell's method with a smooth objective function. Thirdly. for problems in which there are either constraints on the variables or noise in the objective function evaluation. the Complex method can perfOrm satisfactorily in locating local optima. Thus. an efficient search algorithm can be structured within the component to fit the characteristics of the problem and the simulation 62 model being studied. In short. the Complex-Powell combination search algorithm is an effective optimization procedure that can be "tuned" to be efficient for most problems of interest. Subroutines FUNC and TRANSFE The subroutine FUNC and TRANSFE perform the necessary calculations and subroutine calls to evaluate the objective function J'for a given set of input variable values with (see Chapter II-Problem Formulation) J.[ 2,0 En Te 29 2]: M(XL‘.’ B: L t)! mt. 2) - t t 4' Ate: 2; N(z(£, 2. ltu+ 3m“). 2(1‘tu)o ltd} 3m“ NST . = Ppenalty 'Pmult.i 3-9 3.13 the set of parameters of the model. ) where g.is the set of control variables of the model. [0.T] the time horizon. p.1s a set of variables capable of specifying the form.of J‘ needed by the user. A g the set of constraints on the variables. _q (x. _u_. T. y. g) 20. Ppenalty a penalty parameter added to the objective function each time one of the constraints is violated Pmlt.1=ab°. where e is the amount by which the i'th of ncons'r constraints is violated. a.b chosen by the user to reflect the type of constraint desired either hard or soft. and 0 otherwise tn '1' "F [‘57] L" [‘3'] 63 T \_i_(t) —- 3% for (k-—1)tuL SET SHE! SIMULATION PARAMETE! TO INDICATE OPTIMIZATION OPTIMAL CONTROL PROBLEM Alias. mum 2 J~ G p 14y] 3 yi CALL TRANSFB TO PROPRLI ASS IGN ARRAYS FOR INPUT VARIABLE VALUES A CALL SIM TO CALCULATE MODEL OUTPUTS AND .2: f... IF NECESARY W CALLED! TO CALCULATE P13 A“ [(1001.02) W 831' J. P1. P2 F is A mmmm. nmoxmnon mamas: J VII-S CALL smsw I, F ADD PM” PCB (DNS‘I'RAINT VIOLATIONS 4' “2%“ ' Pmum-.1 Figure III-7 FLOW CHART OF SUEIOUTINE FUNC 05 Subroutines SW To insure compatibility between the Complex method and Powell's method. and to obtain a general idea of the behavior of the objective function. the subroutines 8m and SEQSW provide the component with the capability of calculating a local or global polynomial approxima- tion to the objective function. To accomplish this. 31mm czgfiiates a least squares linear regression estimate of the coefficients of the approximation. This regression is performed sequentially in the subroutine meaning that new estimates are calculated from the old estimates and the most recently generated data set. In this manner it is not necessary to retain previously generated data sets in the computer. Calculation of the sequential estimates is rapid while not placing an excessive burden on core storage. For a problem with search variables x1. x2. ....xn and a quadratic approximating polynomial. the coefficients to be estimated are A 3" (a1. a2. Una”) where M: n 3'1 e Znel and. the approximating polynomial _i_s. given by Mat}. 2:2. 00-91"): 2 Inga—1} 391+ 3: Zn afllflfl’fim xix5+ 22: .11‘ +Im+h+l ifl 1-2 031<1 1&1]. 3-11 The coefficients (a1......au) are estimated sequentially as7 a” = a“ * 2r 2:- (Ir-15? a”) 3-12 where 2k = g 3k e 91: 3-43 ith 2 2 2 w 3‘: (X1 pXZ'eee'xnphxl .xleprxszuxi QXungeeexnxn-1 'x1x2eee. "n-i) 66 yk the value of the objective function with input variable values x1. x2. ....xn and 2k the residual error between the polynomial approximation and the objective function values. In addition gr is calculated sequentially from 13r = Er-l'Zr-1Er (“93 131M -1 3:3.-1 3‘” = 10.000 I and I_= an MxM identity matrix. . Er) with 30 = Q and £0 The subroutine SEQSW is an interface subroutine between the component and the subroutine SEQREG. SEQSW initializes the parameter estimates 2, and the matrix 2, and also calculates the vectors 3, Following this a call is made to SEQREG which performs the regression computation. The capability of computing a quadratic polynomial approximation to the objective function is necessary in the component for two prin- ciple reasons. First. the method developed for insuring compatibility between the Complex method and Powell's method is based on an availa- ble quadratic approximation to the objective function to provide a set of vectors that is an estimate of the set of’mutually‘gfconjugate vectors at the optimum. Therefore. this estimated set of vectors can be used to initialize the search directions of Powell's method. In this manner. the relatively small additional computation time neces- sary to provide this capability reduces the number of simulation runs needed to achieve proper convergence of the component. Second. it is important in many situations for the user to have either a local or global approximation of the objective functionu'ZI. 'With this approx- imation. the user can obtain a "feel” for the models behavior or iso- late regions in the variable space that seem interesting for future 67 investigation. The local quadratic polynomial approximation in the variables x1. x2. ....xh calculated during the execution of the Complex method can be described as an n'th degree polynomial p(x1......xn). (see 3-11) Therefore the nxn matrix Q where qii : .1;L :Zleeeeepn 3-15 1-1 1-2) and 2 ’ Q13 1 2 j.“ 1'3 = 10 2.....n. 1*.) is the Hessian of the approximating polynomial and therefore an approx- imation to the Hessian of the objective function. The construction of a vector set that is mutually Q conjugate is performed with a generalized Gram-Schmidt Orthogonolization procedurez. Let = (0. 0.....1. 0.....0). with a 1 only in the i'th t 21L position and zeros elsewhere. be‘a set of n. ixn. linearly independent vectors and Q_be a known symmetric positive definite matrix. It 21 = :1. #.1 = 2k.1.. :1} W323} for"h2. 3. eeepfl‘l and t e“ = —i-¥fl-—333 1:j_<_k= 2, 3......n-1 3-16 .j 923 then 3:32“ - 0 andgiggl 3 Ofor 1:1. m_<_n and lint. These calculations are performed in the subroutine SWITCH resulting in a set of vector directions that can be used to initialize Powell's method. Subroutine SPEED As discussed in Chapter II. Powell's method converges rapidly in the vicinity of a local optimum when the objective function can be closely approximated by a quadratic polynomial. In parameter estima- tion and policy decision problems with complex models. however. very 68 sharp ridges and valleys in the objective function can easily produce difficulties in obtaining proper convergence. The following algorithm. a version of a pattern search superimposed on Powell's method. has been developed to aid in coping with this problem. The basic idea involves finding three points along the sharp valley that straddle the true optimum. A quadratic in the most sensitive variable is then fit to these points. Next. analytic minimization of the quadratic yields the value of the most sensitive variable that is close to its value at the true optimum. A search over the remaining variables. with the most sensitive variable fixed at the value that minimizes the quadratic. results in a point in the full variable space which is then close to the true optimum. Essentially this algorithm produces significant movement along a sharp valley by performing a sequence of searches whose solutions rapidly converge to the valley. The algorithm proceeds as follows: 1. Determine the sensitivities 3x1 of J [xv x2......x’;| at the presumed optimum xgpt -- (x1.0pt' 4,0pt"‘°’xh.0pt) 2. Find sxm such that 1342'.qu for ig‘m. i=1. 2.....n 3. Conduct a linear search on g in y where gaR -—>R and 8(y) = min J[x1. x2.....xm-1. y. M1,”..X'J X l ifm This involves repeated optimization of J in the n-1 variables :1. Xzeeeegxm_1o “laeeeexn With an mad at Vllues HG” xi’opt. resulting in y‘ such that g(y*)5_ g(y) for yaR e ,_ a a e ’4. camato X - (xi’optpe0004_1.opt0 l" O xlmI'optsoeexn'opt) where g(y*) = J (x’). This involves an Optimization of J in the V‘riables X1. Xe'eee'XM-le “IIOO'Oxn with xn=y‘ 69 The point xI is taken to be the local Optimum of J along the sharp valley. If in step 3. g(y) does not have a pronounced. or detectable minimum. then this might indicate that J does not have a unique local optimum. In a parameter estimation problem. this would mean there are dependent parameter subsets in the model. or that the data are not adequate to identify the true parameter values. In a policy decision problem. if J does not have a unique Optimum. then there could be tradeoffs in the input variables resulting in significant savings in real world costs while retaining the desired system behavior. or the original problem could have not been prOperly stated. To complete the discussion of this algorithm. the method.of conducting the linear search of g(y) in step 3 will be described. 1. Bracket the minimum of g with three points y1. y2. y3. such that g(y1)2g(y2) and g(y2)ss(y3) 2. Calculate the value Of y. y’. that minimizes a quadratic pass- ing through the points yi. g(y1)3 i=1.) . As can easily be demonstrated: (yg-yg) g(yi) r (vii-y?) g(yz) + (vi-YE) g(y3) (ya-y3) g(yl) + (y3-y1) g(y25 + (vi-yz) $63) a y :- In the component. this procedure of increasing the convergence rate of Powell's method is used only at a presumed global optimum Of the Objective function. Subroutine SPEED increases the probability of prOper and rapid convergence Of Powell's method even if the objective function cannot be adequately approximated by a quadratic polynomial. Subroutine G§Q§§ The subroutine GLOBE is an algorithm that generates initial points in the variable space for conducting local searches. This is included 70 to find all the local Optima of the Objective function in the region Of interest. Fbr an n dimensional problem the initial points are computed as: x1 = BLi e R1s(BUi-BL1). i = 1. 2.....n where 5': (x1. x2.....xn) is the initial point in the variable space 8L1. BU1 are the lower and upper bounds fer the i'th variable. These bounds serve to define a region in which the behav- ior of the model is to be investigated. and R1 is a random variable uniformly distributed on [0.1] ‘With this subroutine it is also possible to externally specify initial points fbr local searches through interaction with the compo- nent. If a sufficient number of initial points. generated from.this subroutine. are used for beginning local searches. then there is a high probability that all the local Optima of the Objective function will have been found by the componentz. Subroutine COMP While solving a parameter estimationor policy decision problem with the component. it is important to be able to Obtain a running estimate of the various computer costs that have accumulated during execution. These costs accrue from central processor time. core stor- age use. peripheral processor time. and teletype connect time. 'When a large simulation is being studied it is easy to quickly accumulate large computer costs during the process Of solving Optimi- zation problems. TO reduce the possibility Of unwarrented costs accumulating and to provide to the user cost infbrmation useful in a cost/benefit study Of component execution. the subroutine COMP was 71 developed to calculate running totals of the various computer costs. ‘Within the component. COMP is called into execution after each local and global search. Although this subroutine was designed to execute on the CDC 6500 computer. it is possible to develOp a similar subroutine fbr virtually any computer. COMP refers to the CDC system subroutine cpsur to perfOrm the necessary cost calculations. Subroutine LINAPP In a paramerer estimation problem it is desirable to have an ap- proximation to the variance Of the error between the Optimization solution and the true problem solution. If available. this variance of the parameter estimates can provide a measure Of the confidence with which the parameter values can be used in the model. Since the true problem solution and statistics of the component solution are never known in a real world problem. it is impossible to directly calculate this error variance. However. if it is assumed that there is no significant model error and that the model can be replaced by a linear approximation of the outputs in terms of the parameters. then an estimate of the error variance can be Obtained. If (1) there are M output values of the simulation. yl. yz....yn. in the time horizon [0.T]. (2) §f=(x;.....x;) is the set of parameter values that result in the closest fit between the model outputs and the real world data. and (3) f (x). i=1. 2.....M are the functions 3 that map the parameter values x.to the M output values. then with: -P - q — . _I- - f l " y1.b1 o I’ll-13(5). B- 11 X1 0 O * 3M §M_ xn xn 72 where a = Eif 0 id :331 ized model equation fbr the simulation in the neighborhood of the and (1)1 u FEE e g is the linear- ‘X'r-X : : ‘13 ”best” parameter estimates 5?. If it is also assumed that the characteristics of the noise term n is such that Variance (1 given _x) =4?! r 2 1 where 1? b1 b2 <:) 2 or that n is an Mxi vector of random variables distributed with the i'th component having zero mean and standard deviation b1. then the best linear unbiased estimator (BLUE) Of §_has a variance of27 we) = e2(_X_t 1 3.)“- This estimate of the variance Of the problem solution error is calculated in the subroutine LINAPP. Although this estimate is very approximate. generally being correct by a factor of 2. it is. however. useful in obtaining an idea of the sensitivities Of the parameter estimates with respect to changes in real world data noise character- istics. This is infbrmation critical to the determination of the reliability of the parameter estimates to produce real world behavior with the model. User Supplied Subroutines. SIM. NNN. and MMM In a parameter estimation problem. it is only necessary fer the user to supply the simulation model in the required fbrm as the com- ponent will automatically calculate a weighted sum of squares error 73 criterion fbr the problem. However. in a policy decision problem it is also necessary to supply the subroutines NNN and MMM used to speci- fy the fbrm of the Objective function J. In order to correctly interact with and guarantee prOper data flows within the component. it is important that the simulation model be in a specific form. It is assumed that the simulation was con- structed using the SIMEX1 FORTRAN executive programal. This type of program was selected for its general fbrmat to program dynamic simula- tion models. This executive routine (1) provides an organized fermat fOr constructing the simulation. (2) provides for convenient output of the simulation. and (3) permits multiple runs of the simulation ‘with different values assigned to selected model variables on each run. With a simulation model already programmed in SIMEXl. it is easy to adapt the model fOr use in the component. In addition. the adapted simulation retains all the capabilities of the original simulation. Table III-2 contains an annotated computer program having the proper fOrm for the component (*cards represent additions to the SIMEXI executive routine). As can be seen from the program. no re- programming of the original simulation is necessary beyond the inser- tion of 32+ MOUTS e NVAR cards. with MOUTS the number Of model outputs to be studied. and NVAR the number of input variables. The simulation model in this fOrm will properly interact with both the pattern recog- nition and Optimization sub-components. The user supplied subroutines NNN and MMM specify the form fbr the objective function in a policy decision problem‘where ' T J [3. g. ‘1‘] = P1(1(T). T. 2) + j; ”(103). 2(8). 8. g)ds 3:: 74 Table III-2 Adaptation of 81MB! Simulation can mam man swarm - I!" 1013an To sum 1 svaaoc'm's SIM(XIN.COUT.Z.TD.P.INT.IPLO'I‘,TSAVS, ethere- cards 2 h'TS.NSEPJSVITCH.WAR.MOL"I‘S.?!POBJ.P2.NTIMESJS) oeonvst th- 3 0111351011 11::(1).xou'r(1).P(1).'rsm-:(ms.1) uimlation “ RF“ case my." into 5 m9" SIOUX/e ea . “wanting f; comm/moon]... s comps/swore]... 9 DATA mums/"J 10 DATA NVAR/no] 11 IF(IS‘~:JITCH.EQ.1) 00 1'0 10001 0 12 READ 900. may 13 900 warm) 1:» 10001 IFH meow; «o 958 Ex >25 + a a. c - - =1 .. H Jxl i .=. Q! 22.1 1... + J3 443a , C _ {ill—t a. C has? .aetae a. a: 0» . + Am G Owstu + b. as. ovens 0.3 + 8‘ _l-.-Afi T A hes : , 5 sea P name... 3 .1 casu- Oasw % T .589.» , Rial! w 97:2}: \_ wise as F: «0.1:! C / + 023 oz clue . , , fi 9 3526...: xcqun ulllliwav. c _ aofioeoafi + use + + m5 82 commodity: GINV. the size of the government inventory; GIMP. the rate at which the imports are received: PLGIMP. the amount of outstanding orders fbr the commodity that have not been filled by the world market: and GP. the rate at which the government purchases or sells the commod- ity in the market. These six variables represent measurable quantities in a real economic system. Clearly. in a real problem. one cannot expect perfect measurements of these variables. However. to test the component. errors in measurement have been modelled with random variables as discussed in the following section on the parameter esti- mation problem. In the control or policy decision problem the variable TGCOST. the total government cost resulting from.regulation. becomes important. This variable measures the government costs accrued from three sources: (1) government purchases or sales in the private market. (2) govern- ment purchases of imports from the world market. and (3) costs of holding inventory. The cost functional constructed for the example of the use of the Optimization component in a policy decision problem depends heavily on this variable. The time horizon fer the model is one year with a DT step of .05. This DT produces no computational instabilities in the model equations throughout the expected range Of parameter and control variable values. In addition. decreasing DT to .01 changes the values of the output variables and cost functional by only a small percentage. The remainder of this chapter describes the procedures and results for tests on the optimization component using this model. 33 Parameterggstimation Test Problems To test the efficiency of the component in solving a parameter estimation problem. various test problems were constructed. These problems involved the use of corrupted model outputs as real world data. In effect. the real world data was modelled as a sum of model outputs and a random noise component. This was accomplished by run- ning the simulation for a one year time horizon with DT=.05 and at a base parameter set of values. 23133 .-.- (PD. DELI. 01. DEMC. SUPC. KRI. KPI. KP. 1mm” This resulted in 120 data points P(jDT). GP(jDT). GIMP(jDT). DEM(jDT). GINV(jDT). and PLGIMP(jDT) for j=1. 2.....19.20. correspond— ing t° EBASE' To model the measurement errors in the real world data. a noise factor was added to the model outputs. If 013 = P(jDT). 02$ = GP(JDT). 033 = GIMP(jDT). 0,43 = DEM(jDT). 053 = GINV(jDT). 063 = PLOIMPdoo o>owsu< Om oooooz mooaumsam>m dowuodom hm manuauoma< noumom mo someumoaou NI>H maan 96 execution time of SEQSW. SEOREG. and SWITCH depends primarily on n. the size of the variable search set. Thus with a larger simulation. less time is relatively spent calculating the ap- proximating polynomial for a given size of search variable set. 4. From the results in Table IV-3. showing the dependence of the number of function evaluations. N. needed fOr convergence n. the size of the variable search set. a function of the type N(n)=anb. where ar8.5 and b=2 can be shown to give a good fit with the tabulated data. In connection with point 3. above. an approximation can be found fOr the cutoff execution time Of a simulation that determines when the conjugate vector switching should be used with a smooth objective func- tion in preference to regular switching between the Complex method and Powell's method. TO accomplish this it can be assumed that in an n- variable search. the Cemplex-Powell method with conjugate vector switching results in a fixed percentage of savings in computer simula- tion runs. .003 na 1.5(‘Lé—J- “”1) SIM.- '15 (805n ) 4-6 the procedure utilizing compatibility between the Complex method and Powell's method should result in a net CPU time savings in computing the component solution Over the method incorporating only regular switching. Values of 0: and b. and functions for f and N have been replaced by approximations for the use of the Optimization component with a relatively smooth objective function. Test 3: Data Smoothing Capability One of the requirements developed in Chapter II stated that the component must be able to find good parameter estimates in the pres- ence of significant noise in the real world time series. In the first test of the component. good parameter estimates were Obtained in two cases with the corrupting noise having standard deviations of 61:.02. i=1. 2. 3. 5. 6 and c::.05. The solutions were produced by minimizing a normalized. 8:1. least squares error criterion between the real world and model time series. Here it is demonstrated that the 99 component can sometimes provide better parameter estimates in the presence Of noise than those provided by the techniques employed in Test 1. This section examines the efficiency of the component to im- prove problem solutions by minimizing a normalized error criterion between the smoothed real world time series and the model outputs. The comparisons of the accuracy of the problem solutions were conducted fOr three cases. Each case incorporated a different set of real world data generated from model outputs and an additive noise factor. In all three cases the corrupting noise had standard devia- tions of di:.10. i=1.....6. Also. the pattern recognition sub-compo- nent was used to matchthe general curvature of the real world time series with the degree of the polynomial filter resulting in m=2 fOr P. m=2 for GP. 11:1 for 0114p. m=1 for out. 1:1 for cm. and m=1 for PLGIMP (see Chapter III). To study the effects of the smoothing two search variable sets. {KP. KR} and {01. SUPC}. were used in the test ing. The first case used real world time series generated from EBASE : BEASE. This parameter set resulted in model outputs that changed slowly compared to the length of the sampling interval. Again uncor- related noise having standard deviations of g?=.10 was used to corrupt the model outputs. Table IV-5 shows that in this case a significant improvement in the problem solutions were Obtained with the polynomial filtering. Improvements of 50$ for KP. .Sfi fOr KR. 50$ fOr SUPC and 11% for C1 were obtained. The final case that was tested attempted to determine if the smoothing capability of the component could be used to improve the accuracy of the problem solutions when the noise corrupting the time 100 series was autocorrelated in time. As in the second case g?=.19.and EBASE were chosen to generate the real world data. In addition. the noise was generated using a correlation coefficient of'x=0.5 to model autocorrelation in the time series. In this case. it was found neces- sary to use the 8&0 error criterion to produce reasonable problem solutions because the autocorrelated time series had the identical effect of introducing model error in the simulation. Table IV-b shows that in this case the smoothing of the time series did not improve the accuracy of the problem solution. In summary. if it is assumed that there is little or no model error and that the behavior of the system produces slowly changing variables compared to the length of the sampling interval. then it is beneficial to use the smoothing capability of the component to elimi- nate uncorrelated noise factors in the time series. In this case. good parameter estimates can be obtained even with significant noise in the real world time series. 'To determine which order of the poly- nomial filter should be used in the smoothing. the pattern recognition sub-component should be employed to examine the real world model and smoothed time series. Test #3 Powell's Speed-Up Algorithm Although an algorithm that will efficiently find a local optimum along an extremely sharp valley or ridge has not yet been developed. Powell's speed-up method presented in Chapter IV can help to solve this problem in low-dimensioned variable spaces. In addition, this method can only be employed when the major interaction in the varia- bles have been isolated. However. this is an area in which consider— able research will have to be conducted to obtain a satisfhctory 101 Table IV-h Results of Tests With a 3:1 Error Criterion of the Smoothing Capability of the Component Using Real World Time Series 2 5...”.th With 03.9: , Uncorrelated Noise. and EBASEZESASE Component Solutions ' Search Variable Set Using Real World Time Series Using Smoothed Real World Time Series (KP. In) (b.0022. 0.1990) (0.818. 0.203) {C1. sure) (1.997. 1.011) (1.936. 1.101) Table IV-5 Results of Tests With a ~(=1 Error Criterion of the Smoothivg Capability of the Component Using Real World. Time Series 2 1 Generated With 6.0- . Uncorrelated Noise. and p 58-2 E Component Solutions 1 Search Variable Set Using Real World Time Series Using Smoothed Real World Time Series (III. In) Mew. .000502) (#9222. .ooosou) (c1. sure) (1.96666. 1.00250) (1.98821. 1.00115) Table IV-6 Results of Tests With a K=O Error Criterion of the Smoothing Capability of the Component Using Real World Time Series 2 ngerated With 91:9; . =O.5 Autocorrelated Noise. and EBASE= 23135. Component Solutions Search Variable Set Using Real World Time Series Using Saoothed Real World Time Series {”9 n, (.“l732. em) (0“‘30 COWS) (c1. SUPC) ‘100‘9' ‘03,) (‘em. ‘ey‘h) 102 solution. The idea upon which the speed-up algorithm is based has been tested only in one case involving a parameter estimation problem in a model of the Nigerian Cattle Industry1o. As a result of very strong interaction between some of the variables in this model the Nigerian Cattle model was an appropriate test case. The variable search set on which the algorithm was tested involved three parameters CBT, CDT. and C3. In the model CBT is a multiplicative factor relating a general form of birth rate function to a specific application. Similarly. CDT is a factor relating a general death rate function to the application. CB is a parameter specifying the yield of an acre of land in total digestible nutrients for the cattle. The real world time series in- cluded both male and female cattle populations and sales in Nigeria. In this case there were ten data points for each time series this giving a total of forty data points. 'With a base parameter set of E = (1.0. 1.0. 156.)BASE the real world time series was generated from the model outputs (CBT. CUP. (3)3“S corresponding to the base values and with an additive uncorrelated zero-mean gaussian noise component having a normalized standard devia- tion of .10. Applying Powell's optimization method to minimizing a normalized least squares error criterion between the real world time series and the model outputs resulted in. for one set of real world data. an optimum of the error criterion at (car. car. ca) 41.13». .949. 97.37) found in 13“ function evaluations. The final value of the error criterion corresponding to this optimum was 0.63563. 103 Table IV-7 presents the results of three function optimizations performed on the error criterion using three different fixed values of the most sensitive variable CBT=1.0. 1.1. and 1.2 and a search variable set {CDT. C3}. A quadratic was then fitted in the independent variable CST at the values 1.0. 1.1. and 1.2 with respective function values of .bflObB. .63b04. and .63693 obtained from Table IV-7. This quadratic is q(CBT) = .64063-.0459(CBT-1.)e.27u(CBT-1.1)(CBT-1.) 4-7 Differentiating q and setting -agaTP=0 resulted in an Optimum value of q at CBT=1.13H. Then a search was conducted in {CDT. CB} with CBT=1.13#'which resulted in (CBT. CDT. CB) = (1.134. .949. .9737). This entire process required 23031+35o21=110 function evaluations. This is a savings of approximately 22% in the number of simulation runs needed to find the Optimum.of the normalized error criterion as compared to the direct search in {CBT. cm. c3} . Although this is only one test. this general principle of speeding Powell's method along a sharp ridge can be applied in other similar problems. The main difficulty exists in higher dimensional spaces where interactions cannot be easily isolated. Other approaches to solving the problem include using Davidon's method or Hooke-Jeeves search with a very small step size to move a- long the sharp valley. IMuch further investigation is needed. however. in this area. Interaction Capabilities The interaction capability of the component gives the user the potential to select a wide variety of types of optimization executions. Throughout this thesis the various options available to the user have been described. Specific options can be obtained through the input of 1014 Table IV-7 Results of Optimizations to Test Powell's Speed-Up Modification With the Nigerian Cattle Model Fixed Value ef CST 1.0 1.1 Search Variable Set (cm. 03} Values of Search Variables After 2 Iterations of revell'e Method (1.160. 210.9) (.9816. 108.0) (.9136. 78-66) Cerrespending Corresponding Function Value .6h063 e6“ .0693 m of Motion Italuatiens 23 31 35 105 variable-values to the component from a teletype terminal. Although it would be appropriate for potential users of the component. model builders or policy analysts. to discuss the usefulness of the inter- active capability in the program. it is worthwhile to point out some of the important results obtained from incorporating interaction in the component1 (1) Interaction allows for the user to select promising areas of the variable space for further investigation. (2) The algorithm used for the optimization can be rapidly ”tuned" to be efficient through on-line examination of the convergence properties with the objective function being studied. (3) ‘Without wasting function evaluations or simulation runs the user can halt searches in unpromising areas of investigation thus saving considerable amounts of CPU time. (4) From the output of the optimization algorithms it is possible in many cases. to reduce the size of the variable search set while still finding the optimum over the full variable set. (5) The user can obtain a valuable intuitive feeling for trade- offs in the constraint levels on the search variables and different objective functions. In summary. the interactive capabilities allow the user to effec- tively solve optimization problems by properly structuring the search algorithm. investigating the variable space. and examining the effects of changes in the objective function. Poligy Decision Test Problun To test the capacity of the interactive Optimization component to solve a policy decision problem. a realistic problem was constructed 106 using the economic model discussed in the beginning of this chapter. This problem involved the calculation of discrete time functions for the input control variables GP(iDT). the government purchasing rate. and GIMPO(1DT). the government import order rate. for i=1.....20 to minimize the cost functional J erlTGCOST(1 year)+p2P(1 year)(GINVD—GINV(1 year)) . f lmeax(P-PD.O.) .pummPD-m.) B epsmax(GINV-GINVD. 0. )2 epbmax(GINVD-GINV . 0. )2 ep7min(GIMPO.O.)]ds 4-8 where p3>pu. p6> p5. and p1> 0. for i=1.....7. This cost functional was used for several reasons. First. the social benefit of stable and low prices is given importance in the cost functional by penalizing deviations of the price above and below the desired level. Second. importance is also placed on having large and stable inventories. Third. since the time horizon of one year is only a sub-interval for the operation of the actual economic system. it is important to include a penalty for a final value of the inven- tory not close to the desired level. Last. the total government cost of price and inventory regulation is included to account for the government purchase costs. the holding costs. and the transaction costs. To perform the minimization of J with the component. the initial values of the time functions were taken to be the values of GP and GIMPO that result from feedback control of P and GINV with KP=#.O. KR=0.2. KPIr1.0. KRI=1.0. SUPC:1.0. DEMCr1.0. C1=2.0. PDr90.0. DELIrO.4. and GINVD=50.0. Denote the output values corresponding to 107 the feedback control. GPNOM(.051) and GDiPONOM(.OSi). for 1:1,...20. For three reasons. series expansions around the nominal trajectory in polynomials and sine-cosines in time were chosen to represent the controls GP(t) and GDiP0(t). First. the nominal trajectory of GP and GIMPO controlled the price and inventory levels reasonably well. Second. with government control it is difficult to obtain rapid changs in control variables owing to time lags in information 100ps and poli— cy changes. Last. a polynomial of Fourier expansion of sufficiently high degree can adequately approximate any function satisfying the Dirichlet conditions . Now let the controls GPl' GIMPOZ. GPZ. Nnd GIMPO2 be given by the following expansions 1 11 J op1(.051) -.- 33:0 a113(.05i) + opNOH(.051) m 1 GIMPQ (.051) = 2: a12 (.051) 0 GIMPO (.051) 1 3:0 J NON 12 “1 GP2(.O5i) = D a213cos(.051j ) e Z b21j sin(.05ij ) j=0 j=1 u‘2 n2 GIMP02(.051) = X a223c0s(.05ij ) e Z bzzjsinbOSij ) a=a 1:1 for i=1'eee.20 and 352:9 initially. With =1e =20e 3 0e =20. . = 0e p1 3 p3 0 p“ 1 9 p6 . and p7 10 Table IV'b contains the component solutions resulting from minimizing J over 3 and b for selected values of 11. 12. m1. m2. n1. 212:0. 1. 2.... until the improvement in J from increasing l. m and g by 1 was less than 5%. Let 1;. 1;. mi. mg. n;. n; and 3;. 5;. 2. denote the final optimal values of 11. 12. m1. m2. n1. n2. 31. £2. and b. The plots of P. GINV. 108 Table IV—8 Results of Series Representation of Optimal Controls mm RMEENTATIOI 11111111 points 50:20.9. .Variable Search Set 1. 2. 3. lb. 5. 6. 7. “210' {‘210' ‘220’ ‘220 ‘211' } l"4211 {‘210' ‘220' ‘221 5221} {‘210' ‘211' t’211 ‘220' ‘221’ b221 } “210’ ‘212' b 9210' 10 {'210' ‘212' ”212' '220 "221' ‘222 ‘221' ”222} ‘211' ‘220' .2220 5222} ‘211 t’211 212' ‘220’ ‘221 "211 {‘210' ‘211' b211 ‘212' b212' '213 l'213' ‘220' ‘22 h2221' ll222' b22; ‘223' 19223 ro‘%50fi 12"‘1"‘z"’2 0. 0. 19 lo 0. 0. 1. 1. 2. 2. 0. 0. 2. 2. 3. 3- 0. 0. 1. 1o 0. 2. 2. 3. 0 0 e. b; '0‘0029. 30W “00,1190 --15903 41.02377. 3.63230 «611111 . 15.05939 5.267%: ‘3027567 5.8% p -em 4 0 5””. “9 m9 “em: p -3.623h6 -9.1'!080. JAMS“? 10.00016. 0.81669 7.27939. 16.05016 '06“. 3e”” MW“). 4.52735 .1103”. .0335? 4.677780 4.75% 2-6593}. M09230 8.32871. 3-7756? 3.9090. 4.100280 $7095. ~96?“ «759$. -2-33306 $5165. 1.961!” 08$”. 4.5%“ 1.90287. 3.82% M77211. 4.79773 .39688. 1.6101211 3-0279”. .6316! P 969.15 962.67 955-77 959.82 9359.11 957-29 936-% 93".“ m of two "do tar “Cave 25 85 68 2% 262 109 Table IV-8 (cont'd) mm RMfiENTATION Initial points 20:20.22 ’08%5e% Variable Search Set 11. m a; l’ ”or of Mo. "‘1. {or ”Me to {‘100. .120} 0. 0 -emo as” ”9.15 25 2e {‘1103 .111 1. 0 -e“”p -‘el1‘ ”8.82 “2 3e {‘ p I. 0. 1 'eM1p 8.82865 9”e62 77 .32} 12° 46.051115 “a {.1109 .111 1. 1 -ewap -.03755 956.“ 1‘6 ‘120. ‘121} 9e1“mp ‘16s“ so {.100. .111. ‘ 2 2. 1 5.0”. -me6o17 ”2.118 1% .120} ‘1 31.71”. 30;” 6' {‘1100 ‘120' ‘121 o. 2 -.f3”0 90m 60 9”.” 63 .122} ’1 0M9 '01W 70 {01109 ‘11 e 0 20 1 3.0182. 41.213 950.00 260 l l 2 .120. .121} ‘ 22.68530 9e1295 “‘6efiz9 as {C p ‘ 9 C 3. 1 1e8282‘. -.03135 9‘3e1‘. ,1 110 111 112 .113. .120. .121} «em. “e226 9e11,. -16.573'0 9e {.1109 ‘1:1. ‘112 '0. 1 Is“. -1e76291 We“ “17 0 e e I “1907917. 1.0”16 113 11" 120 016.9319 1°e {. p ‘ p . 5. 1. ‘3e9169‘. has”? 939e61 2” afig. ‘11:. a3: -101.i66. 25.fi9 26 26.Wo 25.5391 ‘520"fi;:£ ‘121 ' ' } Om. 10609.13.“ optimal trajectory 110 TGCOST. GP. and GIMPO fOr EFO' gfai. and gfag. bgbf appear in Figures IV-4A. IV-UB. and IV-QC respectively. Note in these plots the significant improvement in the stability Of P and GINV obtained as a result of the higher government costs and final time behavior Of GINV. From the results in Table IV-8. several points should be noted. (1) The coefficients Of the Fourier expansion are more stable than those of the polynomial expansion in t. This probably results from the orthogonality Of the sine-cosine terms. (2) The coefficients converge to their optimal values in the Fburier expansion more rapidly than in the polynomial expansion. (3) For a given number of function evaluations the Fourier expansion results in a lower value of the cost functional J. (a) The Optimal solutions fOr the two expansions are very similar. Thus. a unique local Optimum of J was probably computed. From.an initial value of J=985.56. the cost functional has been reduced to J=934.66. or a 5% reduction in J. With seven terms in the Fourier series the Optimization has succeebd in appreciable reducing the value Of the cost functional and thus improved the system perfOr- mance significantly. If desired. the tradeoffs between different weightings of the factors of J can be tabulated by Obtaining component 7)° This chapter has presented the results of tests conducted on the solutions with different sets Of’ps(p1.....p interactive Optimization component develOped for this thesis. The component can be used for assisting in the solution of parameter esti- mation or policy decision problems in complex models. The next and final chapter will discuss possible further extension of and conclu— sions drawn from the research conducted in this thesis. 111 n. n. ,& m 1 :1 A :0.» D. m 0 00.. o. G 8 .v 0. #0 P G a J 0 o M m o T 5 Am J” T a. a. 0 .0 n 6 .3 o m A 4 2 .m J o A... .J AA see. 0 a. m o a i .6 one 9?. 9H8. 2w? 0.3. a .m: “was . Ewe . 0.8 4 98 came 0.3 98 0.8 93 one 2.8 9% one. on? 9.5 9.3 . .588 98° 6.8.. 0.8... as 6.8? Time fOr a=O Figure IV-hA Graphs of Input and Output Variables vs. 112 (10. & M“ 20.0 ‘400 ‘2. snap R 10.0 0.0 .0 0:1 0:2 * 0.3 ‘ 0.0 0.5 0.8 ‘ 077 1 028 ' of9 '1.'0 yr. TGCOST GIIV .0 50. 199. A.0 we mop; A GIIV 2pm on ”m sf L L me Awe \ -200. % 90.0 0 e p O .1 N 1 O s U 0 e t i O '4 U‘ + 9 at l 0 e fl ' 018 f 039 f 120 yr. Figure IV-hB Graphs of Input and Output Variables vs. Time for 32;; 1153 GIHPO 9.0 or “2-3 A 30.0 20.0 - ‘ -19e0 - «2-0 10. -30.0 0.0 -“0.0 ,0 63.0 100001 59- 01111! L 100. P “9.0 97-5 a 2004.0 “5, 95.0 0,0 07,0 92.5 .0 90.0 400,0 Figure Iv-uc Graphs of Input and Output Variables vs. Time for :22; aux! Effli. CHAPTER V CONCLUSIONS AND AREAS FOR FURTHER RESEARCH The Objective of this research was the development of a computer program that could efficiently solve parameter estimation and policy decision problems in complex models. First. the literature review. initial experimentation. and the fOrmulation of the problem statement led to requirements fOr an efficient Optimization component. Next. techniques to meet the requirements were developed in a two-level interactive Optimization component. The component was constructed in two sections1 a pattern recognition sub-component used primarily fOr examining model behavior. and an Optimization sub-component used fOr perfOrming user specified optimizations. Finally. tests were made Of the validity Of the optimization approach to solve parameter estimation and Optimal control problems. and Of the efficiency Of the component capabilities developed in this thesis to improve the solution accuracy. speed Of convergence. and user costs. The results demonstrated that compared to presently available techniques in certain classes Of prob- lems. the component provides improved problem solutions and improved methods fOr Obtaining solutions. Conclusions The interactive optimization component developed in this thesis has improved both the solution accuracy and the efficiency Of solution for certain classes of parameter estimation and policy decision prob- lems relative tO presently available techniques. Table V-1 summarizes 11h 11n5 e3. 3 so: .23» eaoeu no sea ca usauaoeae “vaganeaao suds eocewaouaee ca sneeze on 0 same» we sea cu usauaoeae yuananaaao saws eocesaouuea peueuuep 1 cameo we see me usauuoeaa hauaaneaeo suds secesaouaea possum-a e .neg suds-ea ease-mead; on.n.u asap-«mes undue-aeuca a one .38.. .305 52.5,. .35... n.3esom «883055 .3: eeeo uaea c« as nepo- pce .mcmnoause newsman suds noose» “aeromuweaoEoo .maz eeeo cue» ca as Hopes pce axeaanou aov e.a~eboa .mdz eeeo ace» c« as Hence one .uca Inuoone Oc .zoaeee moo .eeauee end» pea-Heaaououma menu max eeeo use» ed as Hopes pc- .mca unuoone on .coaeoe auu .neuaee and» peuaaeuaoocs some max ease ueeu a .. 3.... a: £2... .3 .32 mama-m .aoaue on man) dowel deed “diocese £0.33 .70 .3: ceadaenwoo .8 e... p.293 2.2.3» unseen-e0 no seas-zoo mecca-conga I:a«aouue aenpeen- e.aaesom ucamoawse aoaoeo eueusncoo scar anew-ace nonse- aeoenuosa nauseous- aeasueu unweueaue nonse- ae>e~1os9 eedaee end» panes Heeu ed eeaoe peueaeaaeo -3... 5c. Saxon-a senses and» pages «can ad eeaoc povefleauoocs 5s. 2.2.885 coauocpu eeuauenoo essence 0:5 codename esaaoenne sea-see oeeen pecans-Lem e e e e e e e e 0 0 . 0 launch: panacea- ee ease I . n a _h . .n .n .n .h Mi a w MNs um... an u “r m M..m am. h. .m. .. . .t . a 1 1 a 1d 1 e“ I“ 1' ttca drl eh e n sum h 3% eniW m a. «new «an m". Tn» as a...» an i“ no 1d v d n K! [I . .' 1' a“ e a“ e on e o 83."? a nNHhHAHn oaan 116 the results Of the component tests. In this table. the component capabilitites are listed on the left. and brief descriptions of the test cases are listed along the top. The elements Of the table show. for eight component capabilities and seven test cases. whether the perfOrmance of the component was improved. not changed. or degraded relative to the solutions Of the standard problems listed on the far right of the table. Let NLS denote the normalized least squares error criterion. LS the least squares error criterion. and1 pass (PD. DELI. c1. 05110. wpc. nu. KPI. KP. m) (95.0. 1.0. 2.0. 1.0. 1.0. 1.0. 1.0. 0.5. 0.0) (PD. DELI. Cl. DEMO. SUPC. KRI. KPI. KP. KR) (90.0. 0.“. 2.0. 1.0. 1.0. 1.0. 1.0. 0.0. 0.2) 2 28151-3 (PD. DELI. Cl. DEMO. SUPC. KRI. KPI. KP. KR) (90.0. 0e“. 2.0. 1e0. 1e0. IeO. 1.0. 8.0. OeZ) 23.5.; II N the base parameter sets fOr generating the real world time series with the economic test model described in Chapter IV. BéASE' EgASE' and EgASE are parameter sets that produce very low. low. and high frequen- cy responses in the model. respectively. ‘With this notation. the eight tested capabilities. standard problems. and component perfOrm- ance measures used in this table area 1. Estimation of parameters by minimizing a normalized (0:1) least squares. NLS. error criterion between real world time series and model outputs. The standard problem used fOr comparison incorporated a NLS error criterion. the Complex- Powell search algorithm. the economic test model with no erron and real world time series generated fro “Beasaffigass' The accuracy Of the problem solution measured the performance of the component. 2. 117 Estimation of parameters by minimizing a (6:0) least squares. LS. error criterion between real world time series and model Outputs. ‘When tested against themselves. the test case models (1. and 2. described below) were used to compare problem solu- tion accuracy with and without this component capability. In addition. the standard problems were solved by using a NLS error criterion. and the Complex-Powell search algorithm. Eetimation of parameters using smoothed time series. In this case uncorrelated zero-mean noise was used to generate the real world time series. The standard preblemm used a NLS error criterion calculated from the uncorrelated time series without smoothing. and the Complex-Powell search strategy. Again. ‘when tested against themselves. the test case models (3.. 0.. and 5. described below) were used to compare problem solution accuracy with and without this component capability. Estimation of parameters using smoothed time series. However. in this case autocorrelated zero-mean noise was used to gener- ate the real world time series. The standard problem used was identical to the one described in 3.. except that the correla- ted time series were used in the calculation of the NLS error criterion. Use of the Compleerewell search stratengwith regular switch- ing to minimize a NLS error criterion. The standard problem . used Powell's method (or the Complex method with the identical results) to minimize a NLS error criterion. Also. when tested against themselves. the test cases (a. and 5. described below) ‘were used to compare the number of function evaluations needed 7. The 2. 118 fOr convergence with and without this component capability. Use of the Complex-Powell search strategy with conjugate vector switching to minimize a NLS error criterion. The standard problem.was identical to 5.. except that the Complex- Powell search strategy with regular switching was incorporated. PerfOrmance of the component was measured as in 5. Use Of the modification to Powell's method to speed convergence along a sharp valley. The standard problem used the Nigerian Cattle model and the unmodified version Of Powell's method to minimize a NLS error criterion between the model outputs and the generated set of real world data. This problem involved three strongly interacting variables. The number Of function evaluations needed fOr convergence to the problem solution measured the component perfOrmance. Use of the component with the Complex-Powell method to solve various problems of dimension 1 through 10. The standards used fOr comparison were resultsl'a'30 which measured effi- ciency of an optimization routine in terms of the function N(n)-the average number of function evaluations needed for proper convergence in an n dimensional problem. seven test cases include1 Use Of the economic test model with the order of the production and import delays half their normal values to calculate the model outputs for comparison with the real world time series in an error criterion. This converts the test model from a 12'th order model to a 7'th order one. Use of the economic test model to calculate model outputs for 119 comparison in the error criterion. Use of the economic test model to calculate model outputs for comparison. and real world time series generated from EBASE = 1 EBASE° Same test case as 3.. except the real world time series was 2 generated frompBASE - EBASE' Same test case as 3.. except that the real world time series was generated from BBASE = EgASE' Use Of the Nigerian Cattle model. the NLS error criterion. and Powell's method with speed-up modification. Use Of an Objective function that was relatively smooth and well behaved in the variables. In other words there were no extremely strong variable interactions. and no large differ- ences in sensitivities Of the variables. Conclusions drawn from these tests are that (numbers correspond to component capabilities in rows of the table): 1. The accuracy Of solution in a problem with model error is sig- nificantly worsened with the use of a NLS error criterion to estimate parameters. The use of LS error criterion in a problem with model error results in reasonable parameter estimates. These estimates are much better than those found by minimizing a NLS error criterion. From this and the results from 3 and h. a criter- ion for selecting an appropriate criterion is1 0' =1 with an extremely good model. (=0 with a poor model. and 0'8 [0.1] chosen to reflect the expected amount of model error and auto- correlation in the real world time series. 120 There are significant improvements in the solution accuracy when appropriate polynomial filtering is used to smooth real world time series corrupted by an additive. zero-mean. uncor— related noise factor. In addition. improvements are made only when the time series changed slowly compared to the sampling interval. When autocorrelated noise is present in the real world time series. the component provides less accurate parameter esti- mates than in the uncorrelated case. even when special tech- niques are employed to reduce the effects of autocorrelation. In all tested cases Of the use of the two-level Complex-Powell search algorithm. there were significant savings in the number of simulation runs needed to obtain convergence to the problem solution. If the Objective function is relatively well behaved and smooth in the search variables. then the Complex-Powell method incorporating the conjugate vector switching performs more efficiently than the Complex-Powell method with regular switch- ing. Potential savings in the number Of function evaluations needed for prOper convergence can sometimes be realized with the use of Powell's speed-up modification in problems having a small. strongly interacting. variable search set. The number Of function evaluations. or simulation runs. gen- erally needed to Obtain convergence grows at a rate propor- tional to n2. Thus. the component efficiently solves Optimi- zation problems1'3'30. 121 In virtually all test problems examined. the use of extensive human interaction helped to produce efficient solutions of the Optimization problems. This was accomplished by using interaction to eliminate unnecessary investigation of unproductive regions of the variable space. to enable reductions to be made in the size of the variable search sets needed tO locate Optima. and to speed the process Of correctly "tuning" the search methods to increase their convergence rates. Also. the generality of the program allowed flexibility in changing the problem being studied. This should be indicative Of the ease with which the component could be used in different real world problems. TO find the solution of an n-dimensional problem. it is sometimes possible to solve a sequence Of lower dimensional. m TYPE IN M FOR THE TIME SERIES 0002 TYPE IN N FOR THE TIME SERIES GIMP (13) 0001 TYPE IN M FOR THE TIME SERIES DEM (13) 0001 TYPE IN N FOR THE TIME SERIES GINV (13) 0001 TYPE IN N FDR THE TIME SERIES PLGINP aIB) 0001 TYPE YES FOR PRINT PLOT OF TIME SERIES (HE) 0ND TYPE YES FUR PLDT’PRINT 0F MODEL OUTPUT: (HE) ONO TYPE YES FOR SIMEX STUDY OF MODEL (HE) 0ND TYPE YES TO CONTlnuE PATTERN RECOGNITION ETUDY (H63 ONO TYPE YES TD MPITE TRPE98 (RE) OYES .....END OF PPDERRM..... END MRINPR nK-CRTHLUG.TRP293ssxmasaa.ID=30£HEenEP.pD=999. CRTRLOGpTHPEQS-EIM98HPpID=EUCHNERsRP=9?9. IDS QEEIldJuflhhonTSubwikngggnont 01/07/75 MCU HUSTLER _ L335 L-D 36.37 01 03‘75 TYPE PRSSMUFDa PM: HND UIEP ID. [IIIHEIBBI909554+¢49sBUCHHER. INVRLID PROBLEM NO. TYPE PHESUOPD: ON! AND UZER ID. Ill1533833909054499+¢49sBUEHHER. 3356?879 LINE 46 LRST RECESS: V 0110 RUNS: IS BHLRNEE «75 00:05 I 95.51 “ . 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COMPLEX PROCEDUFE OF EOE PRRHMETER: N a 7 M = P K = 3 ITNH= = $00 IC = 0 RLPHN = 1.30 BETH = .00010 38000 = 13 DELTH = ITERHTIONZ. PREZENT MINIMUM. HND PUHETIOJ YHLUE 1:... IT8 30 X: 1.7902 .9503 .-S 3 1.049? 1 13?2 0.98:3 Fa ~13.34932 TYPE IN YES TO CONTINUE COMFLEH ITEPHTIONZ €861 OYE3 ITERHTIONZ. PRESENT MININJN. BNO FUNCTION VHLUE IT... IT8 00 .4 a . 1.?995 .9??4 .95?? 1.0333 1.1091 3.0301 F: -1.S?810 . TYPE IN YES TO CONTINUE COMPLEX ITEPHTIUNC 196) OYES ITERHTIONZ! PPEEENT MINIMUM; HND F:UNC'TIUN YHLUE IE... IT. 90 x- 1.?960 .9837 .9433 1.1033 1.0392 5.6554 F8 -1.09513 TYPE IN YEE TO CONTINUE COMPLEX ITEPHTIDNE (RE) OYES ITERRTIUNS: PRESENT MINIMUM: HND FUNCTION VRLUE IS... 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CDC 6500 FTN V3.00P380 OPTIT 162 -SUBROUTTNE SUBOPT G 9 S R 9 T R T 9 O s G 9 O T E T E E T T H "99.! R N U 99 T 9 9T N '- 0 Y... W .LQYILI . T. X ”MU LIQKY. T T o C AO(T( I C N A( . r¥JL(L J o .IUNH E SUTLT 9 E A E9 H EOHIH T N 0 PC To 9xOH° In. T T 9 9 T ..1L0C . I. ...II P TNICO J P a TT P 91’ 9B 9 p T 58 E LIATS 6 m ( L( T NTLTO ( T T "T T 08((V L c T C 9" P C(HLt I J P . CO C 0I0(0 H o C N 0C 0 v. OCH 0 o T 0 T A T9 T S DH.UY C 5 T S V TT I S O AOTC. ( ( s 0 N 6T J 0 C OCTlh o L 0 C o (7 ( C T 9.3T. 9 ( C T 9 LT S T C T AT TH T T N T C U (( 9 C 9 T 'T(6N 9 T 9 C o s "L T 9 T . O3LTO T c C 0 T N 0( T T S J TT(TS J o T 5 0 CN . s 0 J T(NLT T J T s 0 V 00 J 0 r. . .zLOIU N o .I 0 .L E .LC . C H T T(CHU T T . C N D T. T N C 6 IN .UH R 2 J H C . JI 2 C o 2 LOTCO P T 0 C o T Y T Z 9 'l I. (\g9R T T l! T 9 T To 9 ‘6 T T 5 TL HoNLA T 0 TL TIA T S . 5 LT L 5 0 T( OTUVV . o T( TJ( S 0 TJ 3 o (( ( 9 C TH CTCAN J 9 TN TJL 9 C To 2 5 KL M C P 90 obNSo o E 906 90( C P TT 9 N 0( U P P T TC TToTT T . TC9 TTN P P 92 T E CH C P 0 6 T 0 TST 9T 9 T T 0T TTO P 0 TT 5 0° 0 9 T T .T TRLNSO 2 Z .T ..C o T T T( 0 N 9CT T T 56 JT C(TTZ ( ( JTU JT. T S E J .L T o ION J 5 0T 0 JT 09M0T( L S JTT TT 5 0 8 ( J( T TE. N 0 C T T T o( TTNO.‘ 0L ( T O( 0 OJ 0 C 0 N a" W T NVS ( N C P0 J J TE T TC9J( N N. TEU TTTJ C P L A TO 2 TAT E TT P CT ( 0 ( 8H .ONoGHN 0 TTO BHG 538( P C G V 8C 2 OSU H A0 C Q X G A (A JGFTCUU C AON (A N((E C Q NRA (I T ( "TO 5 A (L 0 T0 8 .. “N T o oTHPL I (.uA LNT ULLM 9 T .A3 LT S L 90H N N 00 T 56 T T T (.T o TTNXNN. T 05. (00 C((A GT 5 LVT (T L ( GS! E o FNEMS 0 T N T NNJN E bOvocT T NEBNHJ. NNHN XS N0 GNTNN. ST N ETT S J EOUTU CT . T . .OJF 8 Z.AHPPT . 08.90J0 9009 EU TC T...0T NH 0 NTT N N 0C. OCT OT NJ H NJ TIC o O 0N(U( 00 99 NJ TC . OTC 9E TCrCT OCT 0 I ITJTCI. 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OTNTOOOTUTUSTTLTTZZUCTO .TNLONP(T HT N TD C0 LE ( 0L (N .LC NLTC(5HPNH (MN TUTTC U TCC N CU G T(I..L( 851...).50NH .CU TTONT CTGISLOFI6LTT6 TTTHTIS DVUEOO9LETTONS9 TCT9 TTT CTTTT9L9 LT N(S T5 00T9$T TNLENLoNTNT(TENT(TNTLNNTUTTTLTNPCTT(sTNbETTLNNNTLLNTNLoNN:NT(TL(NUAHNLNTPCNTNN NTD=TL III M (1 MN1 LTTNL J(L MbNTD=( LTIT LLTNTL TILT H IMTA(ULL.P T :1 ORNHRA HFHUUUFNUUUHUAHRUUU(FA OUUXORNHFUARFROAARURA RRGHUOUAUHEFSSA D RUJH CPETPC PTFUCGTFDCCPOCPPCNDXTC GOCECPETTDCPTPDCCPCPC PPTPDCDCCPHTNNC N PDNP 0 n. 2 2 T a T T T6 T T 2 3 S A 66 6 T T 79 a 9 99 9 9 9 T T T TT T T T TT T T TT T T T 160 165 170 175 180 T05 T90 T95 200 205 210 215 220 225 230 235 260 2‘5 250 255 260 265 270 275 205 290 295 300 305 310 SUOROUTTNE SUBOPT 163 CDC 6500 FTN V3.0'P380 OPT'I 01/27/75 .2: 1923 CONTINUE PRINT 1027 READ $027oAN IrtAN .Eo.3HYES1 GO To 99 PRINT 1028 READ 2028.AN IFIANS.EO.2H HNO) GO to I96 PRINT 1029 READ 20299NCHG DO 193 JaloNCHG PRINT 1030 ‘93 READ 203 .J J.XVARIJJT 9A CONTINUE PRINT 1031 READ 2031.AN IF1ANS.EO.3H HYESTGO TO I PRINT 1032 READ 2032oAN $11ANS.E0.3HYES) GO IO 2 HINT 1033 READ @033.ANS . 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VAL IUE SENSITIVITY 0) 10061 FORMA111«.3A.AB.§F12-31 _ 1007 FURHAII' TYPE YE T0 EPEAT SENS TESTING IAGT'T T887 FURMAIIA61 B FORHAII' TYPE YES TO EAIT FRON PROGRAM (A6191 €008 FURHAIIAGT 2883 :8R::;:;3}YP E IN THE NUMBER OF VARIABLES To BE SEARCHED-N (13101 H 9339 TSSNTTTTST'PE IN THE NUMBER OF 0T UNITS FOR OUTPUT SPACING (IST'T 1010 lZUHMAIIP IYPE YES To INPUI INITIAL POINT FOR THIS GLOBAL SEARCH 1A 0 FURHA (A6 ToTsl FUNNATI' TYP IN INITIAL POINT FOR GLOBAL SEARCH (7F10.ST'T 20101 FORMAT(7FIOP S) 1011 [FUSMAIIP ..POLYNOHIAL COEF. FOR THE LOCAL APPROX. ARE.../IIA.SEIA. 10 Z FORHATIJA.IS.ZA.A6.ZA. F105 0 02 FUHNAT‘. .0... AL UPT "UN ,OUTPUTOOOOO.’ 0 3 fURMATI‘ ...TH E CONSTRAINTS ARE... 0 3T FONHATIZXQAOOZX OFIOC SOZAOFTOOST 0 Q EUNHATt... I’MUH TUNCTIQN LV‘PUE..'ECS.S 0 S FUHMATT° ..ELAPSEU TIME UH THIS LUL .I'oféo 0.5T 0 o [TBHQTT‘. THE PHESENI CUSIS ARE (DOLLARS) TOTAL.CP.PPo HoCT0/128.5F lOITI FUHHATT. ..THE PRESENT MINIH UH OF LOCAL 0PT. IS...) 0 7 FORMATIZX IS.ZA.A6.2A.F10.51 ‘° 3. 05:11:: *"E “25352: 351133: 3%“fl‘éfi' It... .00.. 8 v 1 UNg?:I. ..PULYNOHIAL CUE}. FOR THE GLOBAL APPROX. AHE..0/11xoSEIA T020 FORMAT(3X0T59219A602!9FT0.5T 163$ .suanourlNE 50809! CDC 6500 FTN V3. o-Psao OPY-I 0:121/75 1021 Fgng;(: ;gngEOCOngg?{ELSSGSE;.00/éixGQ6;ZXVF10.592:o210o5)i 315 {353 EUHMQTg' TH LLAS gto TIM F0 $HQ S “GLob L'sL ' EDEN...“ 1025 FURNAtt° 1H NU. ur LocAL StAHCHES TON?UCY§8“ FOR THIS oGLOBAL SEARc 1H=°olax° IN NO. or PNOB ENS XNV 51 GA E 1921 RM Ar(- ..SENS.0A IA AI LuaAL OPIINUNANE VARIABLE N0..NAHE.VALUE. I AND SENSITIVIIY° 320 ‘026 YORHAY‘ZXVIQVZXofib W5 :10. 02X. F 285; :83:2;:;6IYPE YES nun o I 3RO0RAH (A6D'D gggg Egfingggzbivpt YES 70 CHANGE DEFAULT VARIABLE VALUES (A6).) 325 1023 :83::;:;3}Yp£ IN YHE NUMBER or DEFAULI VALUES to BE CHANGED (13:0, 0 Egég F3::2¥:;a‘;?5 IN A VARIABLE No. AND NEN DEFAULI VALUE (lboF10.5)°) 33° 53;} igunzgfzbiVP£°vEs to RETURN to UAtAl (A6).) R #332 :83::{(x6}YPE YES to REtuRN to OAYAZ (A6)°’ 8%; $3332}3A6}YPE YES to REIURN to DAIAJ «Ao)-) 335 $83“ ru:::;:zbtvpe YES to NEIURN to SENS. IESIING (A6)'! Io U . ) 3332 r8:fl:}:£"l“'. Eggabcnsm INVESIIGAYED') 3‘0 1323 :333:}:;,‘3?5"" rue RAIE 88N3u3° kon' rnxs ExecutxON (71.030) €041 FURNAtt' iYPE YES to OBYAIN DATA REYURN AND Ell! OPTIONS (A6).) a a A A6 a RMA - u 11 AL A 5 0A - f§“§ £092.13. évug g 53 Max; 22~51¢E2r1ao sxgz.. 3A5 1051 EggnAt(° IV? to NEIURN to DAtAa (A6 -SUBROUYINE CONV SUBROUYINE CONV(!VIX) I!!! REYURN END CDC 6500 FYN VSoO-PJBO 0PT.! 01/27/75 .22 16H} -SUBROUTTNE SUBOPT CDC 6500 FTN V3.0-9380 OPT-l 01/21/75 1821 :8:HA;:: ;goTHEoCOngg?}=$:FESEDOo./¢2X;gbozgogl0:592‘DF1005’, HA ALU 3.0 315 T053 FUNNATT' THEG ELA 0ggko TIH E "TN ’3” GLUUA L SLA&CHI°VF10 .5) 1025 FURHATT° TN NE NU. ur LOCAL HE $0 NDU CTED FUH TH s fGLDBAL SEARC 1N=-.1A/° THE N . or PRDBtEHb 1NV ST GA to U FARI o 1921 F RHATTO ..SENS.DATA LUBAL OPTIMUHARE VARIABLE NoooNAHEoVALUEo I AND SENSITIVITY.) 320 1026 F0NMATT2A.TA.ax.AS ;SA.F10. ozx 021 FURHATT' TYPE YES NDN o T PROGRAM (A6101 foe? FURHAT(A6* 028 FUHHATT' YPE YES TO CHANGE DEFAULT VARIABLE VALUES (A6).) 2028 FUNNATTAS) 325 1823 :8::2T:;3IYPE IN THE NUHBER 0F DEFAULT VALUES TO BE CHANGED (13).) {030 FUHHAT(' TYPE IN A VARIABLE NO. AND NEH DEFAULT VALUE ([69F10.5)') 2030 UNMATTTAVFI . E 1031 0RHATT° TYPE Y S TO RETURN TO DATA] (A6).) 330 2031 FURHATTAb) #332 :83:2¥(;61YPE YES TO RETURN TO DATAZ (Abi'l 03$ FORMAT!“ TYPE YES TO RETURN TO DATAJ CA6).’ €033 FDRHAT(A6) 335 034 FORMATT’ TYPE YES TO RETURN TO SENS. TESTTNG (A6).) 2039 UNNATTAST , 2035 URMATTZAoTbo° PROBBFHSU80 TNVESTIGATED') 2036 FURHATT. ......END 0P5 00. 0A0 FORMATT' TYPE 1N THE RATEB 6H ufi FDR THIS ExECUTTDN TF1.010T 340 DAD FUNNATTF1.01 §A1 FURMAi(; TYPE YES To OBTAIN DATA RETURN AND EXIT OPTIONS LAST.) A RNA 6 4 80M. . Tu A ua 0A... f0“; EURMATg’ rvfig Ygg TXSK L I; 2% TING 51%;.) 365 1051 E33HAT¢° TYP Y TO RETURN TD ATAA (Ab) oSUBRDUTINE CONV CDC 6500 FTN V3.0-P380 DPT-l suanourxus CONV(ToTI) RETURN END 0‘127/75 .22 IS 20 25 30 35 60 65 50 SS 60 65 70 75 mo Ofl uflI cc 0000 00 C00 00 0°06 6° 00° om mbbu kw ~60 Ov~ ”NP-WU 1(35 DATAI CDC 6500 FTN VJoO'PJDO OPT'I 01/28/75 SUBRUUTINE DATA!(NDPTSOTTSvNAHEoXVAROONAHESoISONTSONVAROHOUTSONDP) DIMENSION NUPTSINTS) 0TT5‘NTSONOPIONAHE‘NVAR)O‘VAR‘NV‘R’O 1 UNAHES(MU T51o1512 0) PWINT.I000 READ (0009‘ NS IE(ANS.E0 HNU) 60 TO I REAOT9a>T§ 11 READI98I READIQBI(NAHEIJIQJIIONVORI REAOI98) READI98IIONAHESIJIf J. 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