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Optimal Control Sectorial Model of the Polish Economy with Flexibility in Criteria and Constraints
Copyright © IFAC Dynamic Modelling and Control of "'ational Economies , Edinburgh. UK. 1989
OPTIMAL CONTROL SECTORIAL MODEL OF THE POLISH ECONOMY WITH FLEXIBILITY IN CRITERIA AND CONSTRAINTS K. Cichocki S\'slems
Rl'Search
Inllilulf ,
Pulish Aradcmv of
SCifll(fS,
Nfll'l'lska 6, 01-447 Warsaw, Poland
Abstract. A nonlinear, sectorial model of the national economy is investigated. It is formulated as a multicriterion, o~timization problem with constraints in the form of difference equations, linear and nonlinear equalities and nonequalities. A scalarization function is a~plied to transform it to a discrete in time non-classical optimal control problem with box constraints on control variables. Flexibility is introduced in a~plying various constraints and various objective functions at the construction of alternative multicriteria optimal control (MOC) models. The objective in the model is to minimize the distance between desired paths (rates of growth of consum~tion, national income or other economic indicators) and the ones obtained as solutions of the model. Computer procedures are elaborated on the IBH PC. They include: main control program, descri~tion of several versions of the model with computation of values of constraints, objectives, state variables and their derivatives, input procedure, several versions of model solutions (dependent on the type of constraints) and the output procedure with graphics. Exemplary solutions are given for the period 1986-1995. Keywords. Modelling of economic systems, o~timal control, multiobjective and nonlinear optimization, com~uter sinulations, computer procedures. INTRODUCTION
periode or at least followed with a minimum distance.
The objective of the paper is two fold: first-to present various versions of a se~ torial model of a national economy, formulated as optimal control problems, and secondly - to describe a package of computer procedures on the IBM PC which describe and solve the above models. Flexibility is introduced in applying various constraints and various objective functions at the construction of the multicriterion optimal control (MOC) model of the economy. Flexibility is also incorporated in the solution procedurs of the nodel. It allows for solving multicriteria or single criterion problems with equality and (or) inequality constraints and (or) box constraints on the control variables. Additional linear difference equations can be included.
The objective of the model is to minimize, subject to constraints, a distance between the desired paths and the ones gene rated by the model. Given the desired paths (or a path) a single solution of the formulated optimal control problem yields the optimal values of control variables which satisfy the constraints and ensure that selected objectives are at the minimum distance from the desired objectives paths. The Eucleudian norm is introduced. The elaborated model belongs to a class of discrete-time optimal control problems with additional upper and lower bounds on the values of control variable s. The multicriterion problem formulated at the outset is transformed to a single criterion problem with an aid of a scalarizing (achievment) functions (Wierzbicki, 1979 and Chmielarz, Ci.c hocki, Stachurski 1982). We have selected various constraints and various objectives in order to construct versions of the MOC model of the national economy. So far eleven versions of the model have been programmed on the IBM PC and investigated.
The model is formulated as a nonlinear dynamic, multicriterion optimization problem with constraints in the form of difference (state) equations and linear and nonlinear equalities and inequalities. It is designed for assisting a deciSion maker in investigations of alternative economic development strategies and for checking the consistency of assumptions of economiC projects and plans. The assumptions concern rates of growth, over time, of basic economic variables which characterize the national economy. These variables include consumption, investment, evmployment, exports, imports and resulting from them - production, fixed assets, net national income and foreign debt. Hypothetical assumptions yield desired paths (rates of growth), for instance of consumption, or of national i~ come, which, in the view of decision I!'akers, should be achieved within a given time
The !lOC model can be applied in practice: firstly, for checking the consequences of a selection of various alternative desired "aths, secondly, for investigating "the inner consistency" of the model, i.e. for checking the feasibility of assum~tions of a simultaneous achievability of a number of desired ~aths, which represent desired growth rates. If the simultaneous achievment of the assumed growth rates is not possible-for instance requires high imports or (and) very high investments, then we conclude
245
246
K. Cichocki
that the assumptions of a development plan are not consistent (it can be true in both cases, of feasibility and infeasibility of solutions). A hypothetical way of utilisation of the MOC model in a decisiona-aiding process, for instance by a decisiona maker in investigating and elaborating an economic plan, is presented in Fig. 1. The ?lanner takes into account the world market situation, home economic conditions (resources, technology and technological progress) and formulates an initial development scenario, based on his knowledge of the capacities of the economy and of expectations of the society. The "initial scenario" is verified in result of its inner inconsistency or due to investigations of a hypothetical development scenario. In the process of confrontations of the planners goals and intentions, expressed by the desired paths of growth and the results yielded by solutions of the model, the planner can change the values of the desired paths or he can choose new goals of the economic developnent. For instance he can either try to satisfy a given consumption level or to diminish the foreign debt, or to attemp to do both at once. Another objective can be the maximization of employment in the economy. Computer procedures in FORTRAN on the mM B:::: have been elaborated. Their purpose is to describe the set of the MOC models of the national economy, enable to solve the model, conveniently change the input data and present the model solutions. The set of procedures c~nsist of: the main control program (MAIN), input procedure (INPUT), procedure of the models description wi~h supporting computational procedures for instance of computing the values of state variables, constraints and of objective (MWP) , the solution method (SOLVER) and the output pr~ cedure (OUTPUT) with suitable graphiCS. THE MOC MODEL OF THE NATIONAL ECON0I1Y Description of the Model The multicriterion optimal control model, with slight simplifications, can be written in the form min f(x(u), u) where f=(f , ... ,f , ... ,fn) , n>l (1) 1 i u and f : RlT x RsT -> Rl , i = 1, ... ,n i
ST U E Un C R . The function f transforms the feasible set into the set of achievable pOints Q since, given u the relaO tion x(u) is uniquely defined by equations (2), (3), QO=f (U D) . The multicriterion problem (1) - (6) is transformed to a single-criterion one with the help of a scalarizing function method (Wierzbicki 1979). Minimization of the scalarizing function J(u,f) substitutes for minimization of n criteria of the form (1). The function J(u,f) is of the form n
2
_
J(u,f) = p1: Gi[max(O,fi-f (x(u) ,u))] i i=l n
- 1: 0 i[fi(x(u) ,u» i=l
(7)
- fi]2
where f. are desired paths (reference-paths)~ which are determined exogeneously, 0 . are weighting coefficients and p >l is a~penatly parameter. _ n Changing the desired path f E R one can obtain any path f EQ . If f is_not achievO able the minimization of J(u,f) corresponds to the minimization of n _ 2 (p-1) 1: c .(f.-f.(x(u),u» i =l ~ ~ ~ while when it is achievable, the minimization J(u,f) corresponds to the maximization n
of 1: (f.(x(u),u) _f)2. i=l ~ Economic Interpretation of the Hodel In the MOC model we define the following variables: control variables, which include the vector of consum?tion C = (C , ... ,C ) , the investment vector t1 tm t Vt = (V t1 ,··· ,V tm ) , exports E t = (E t1 ,··· ,Etm) , ~mporEs for consumpt~on purposes Ht = (/\1' ... ,11 tm ) and employment L = (L t1 , ... ,Lt)' where m is the number ot sectors in ~he economy and t=O,l , ... ,T-l; T denotes the length of the investigated period. state variables which are functions of the control variables include the vector of fixed productive assets Kt = (K t1 , ... ,K ) tm and the scalar value of the fore~gn debt St' defined for the whole economy, where t=O,l , ... ,T.
gj(Xt , Ut) ::; 0
j
1, ... ,p
(4)
gk(ut ) = 0
k
1I
,r
(5)
The linear difference equations describe the behaviour of the state variables over time. There are m equations of ty?e (2), which describe accumulation of fixed assets. At time t+l it deoends on the value of fixed assets at time t decreased by a part of it which is depreciated, and on distributed over time investments. Part of the fixed assets 6 determined over the time (O,T) is due to investments committed prior to time t=O.
t
0,1, ...
(6)
Kt +1 ,j=(1-d jj ) Kt + T: /T J Vt- T,J + 6 tj
subject to constraints N ht (u _ ) , x t +1 = q,x t + 1: T t T T=O x x 0
t=O, 1 ... ,T-l
(3)
0
~t ~ Ut
$
(2)
•••
t
Ut'
Ut > 0;
I
T-l
where n is a number of criteria, ht and gk are l~near functions, g. is a nonlinear function, q, is a constarlt matrix of the dimension 1 x 1; u~ is a vector of control variables, Ut E R , and x is a vector of t state variables,
The controls u belong to a feasible set U , determined by constraints (2) - (6); D
t
(8)
with given initial value j
=
1, ... ,m
( 9)
The values of ~ t. are time dependent coefficients of invJ~tments delays. The accumulation of debt St+l is described by a single difference equat~on which explicitly depends only on past credits (value of the debt at time t), imports and exports.
Optimal COlltrol Sectorial Model of the Polish EcollolllY
m 5t+1=(1+rt)5t+j:1 [GtQtj + (HtVt)j + Mtj - Etjl with given 50 = 50
247 (10)
The value of r t represents the interest where Tl is the liquidat10n time of an agrates on crediEs, while Q , is the value gregate investment, for instance of cont of production of the j-th ~ector derived structions in the chemical industry. If we from the balance equation do not want real decapitalization of fixed m m Qtj - i:1 atjiQti + gtjjQtj + Mtj-Etj-aj(0tj-0t_1 ,j)=i:1 (btji-btji) Vti + Cti ; j=1 , ... ,m; t=O, ... ,T-1 (11) where a" are elements of the intermediate material1coefficients (Leontief) matrix, g" are elements of a diagonal matrix G ofJthe average propensity to import for oroduction purposes, b" are the total and t" imported capital irt~estment coeffients irt1the j-th sector. Ht is a diagonal matrix of the total imEorted investment coefficients h = b . The coefficients a <1, j ji jj
i
determine a part of a production increase which defines the reserves. The production capacities are approximated by a nonlinear constant elasticity of substitution (CES) production function. It is assumed that the production distributed to sectors of the economy can not exceed production capacities (12) where mt' is the coefficient of utilization of Jproduction capacities, O::illltj ;S 1. The consumption in all sectors is greater than a given consumption subsistence leveL The other control variables are assumed to be either nonnegative or lower bounds on their values are assumed. Usually in all but one version of the model full employment in the economy is assumed. The summation of employment over sectors yields a time - variant employment in the economy.
assets we can require as in the 3rd version of the model that (16) The constraints (15), (16) do not show up in the versions 1 and 2 of the model, in which it is required in the objective function that the fixed assets follow given exogeneously desired trajectory. The 11th version of the model allows for unemployment. Its objective is to maximize employment while in another, not included version of the model we can maximize simultaneously employment and consumption. The arrows in Table 1 indicate selected ways of carrying out comparisons between consequent versions of the models. Firstly one can investigate implications of the selection of various goals, which are to be achieved by the economy, and consequences of introducing or relaxing the constraints related to the level of fixed assets, new credits, foreign debt, the national income etc. Secondly, one can select various desired growth rates of the consumption, national income, fixed assets, foreign debt and employment and investigate the achievability of thes e desired paths. In case of selection of several desired paths at a time we can check if a simultaneous achievement of these paths of desired growth rates is possible. If the desired growth rat e s which can represent assumptions of a given economic d e v e lopment project (for instance of the economic plan 1985-1990), can not be achieved simultaneously, then we conclude that the assumptions of the plan are not consistent.
In Table 1 we present eleven versions of the MOC model which have been programmed and investigated. They differ, one from another, in the form of the objective function J(.) (the number of objectives selected) and in the number and (or) form of DESCRIPTI ON OF COMPUTER PROCEDURES the constraints. For instance the 6th verOF THE MODEL sion of the model assumes the objective function of the form J(D,D)in which we are The Input Procedure interested only in the growth path of the net national income Dt , while the 7th verThe procedure allows for introduction of sion allows for invesEigation of the growth suitably prepared sets of data which are rate of both the consumption and the nationecessary to solve the MOC-model. The set nal income paths. In this case the objecof data is divided into more than twenty tive assumes the form T-l m T-1 _ 2 T-l m _ 2 T-l _ 2 _ 2 J(C,D,C,D)= p { 2: l: ,[max (0 'CtJ--{;t ,) 1 + l: OD [max(O,Dt-Dt ) 1 } - { l: ,2: 0c,(Ct'--{;t') + l:0D(Dt-Dt ) } =1 cJ J j J=l J J J t=O t=O t =O t=O p>l (13)
°
where the nett national income m m Dt = l: (Qt' - , l: atjiQti' t=O,l, ... ,T-l j=l J 1=1 i;
(14)
In both versions, 6th and 7th we assume that the yearly increase of the foreign debt (including new credits) should not be greater than 25 per-cent of the total exports at a given year. Additionally, a lower bound on fixed assets is assumed. It results from a linear rate of depreciation (liquidation) of capital, provided no new investments are committed in the investigated period, i.e. o ' Kt , ~_K to ,-K to ,t-t (15) -m---r- + 6 , , J=l, ... ,m tJ J ,J J ~lj-LO
subsets, each including several data units. A single unit constitutes for instance a matrix of dimension 15xl0 whose elements are e.g. desired values of consumption over a ten years period for fifteen sectors of the economy. Another data unit can be a material coefficients matrix of dimension 1 5xl 5. The introduction of the data is organized with a cooperation of the user who selects a data unit in each subset. The size of the matrices and (or) its elements can be changed. New data units can be established. Changing the data of a given data unit is done outside the Input procedure. This procedure will be more user friendly.
248
K. Cichocki
TABLE 1
The comparison of various !10C models
2
3
4
5
J(C,K,S) J(C,K,S,D) J(C,S,D) J(C,S,D) ,J(D)
6
J(D )
8
7
J(C,D)
J(C,D)
9
J(C)
10
J(C)
11 J(L)
K
Kt~tO
K
T6t-t KtO .......................... Kt ->Kto - ~ T -t Kto I 0 I 0
C
S
D
L
l:Ltj = It .......................................................................... l:Ltj= l t l:Ltj ;;lt
/. The Solution Procedure of the Model The so l ution method is elaborated and programmed in the form which enables the solution of a discrete in time, optimal control problems with constraints on the control and the state variables also of the type described by the MOC model. The method is structured in a flexible way . Dependent on a value of a special parameter of ITRYB it solves either a problem with box constraints on the control variab l es only, or a problem with additional inequality constraints (or inequalities and equalities) . Additional possibilities include linear state equations and equality and (or) inequality constraints. The nonlinear, discrete optimal control problem (2) - (7) is solved with the help of the augmented Lagragian method (shifted penalty function method) - (Bertsekas, 1976; Hesteness 1969; Powell 1969; Rockafellar, 1974). The Lagrangian is minimized in the control space. The linear state equations (2) with the initial conditions (3) enable exact determination of the state variables x , given controls u. Thus, the problem can be reduced to the control space, reducing also the dimensionality of the problem. The reduced gradient of the augmented Lagrange functional can be computed and the ajoint variables determined since the linear state equations (2) are explicit linear equality constraints with value s on the state space_ Thus, nonzero multipliers for those con strainte exist . For given " penalty parameters " we are mini mizing with respect to controls u the augmented Lagrange functional subject to the bounds on the control variables (6). The applied minimization bases on the Newton projection method for minimization of twice differentiable functionals subject to box constraints and is taken from Bertsekas (1982). However, in the subspace of free variables we do not use the Newton direc tion . Either a decent direction is used when the active constraints change or a conjugat gradient direction otherwise. The directional minimization util iz es the non-gradient method of Fibonacci. It results from the fact that the functional dependence
of the value of the minimized function and the length of the directional step is not necessarily differentiable , because of the projection. Thus, dependent on the value of the ITRYB parameter we are applying either a method of the conjugate gradient with projection on the box constraints or the augmented Lagrange method with the r educed gradient or the augmented Lagrangian method only . The solution procedure consists of several subprocedures which are shown in Fig.2. The subprocedure I1UST minimizes the Lagrange functional in the control space, utilizing the augmented Lagrangian method and the conjugate method with projection on the box constraints. The reduction of the problem to the control space is initiated or executed in the subprocedure FLAG or in GRAD. The subprocedure GRAD calculates the value of the reduced gradient of the functional in the control space (computes the adjoint variables) . The subprocedure PROJG determines the working set of active constraints in the k -th iteration of the conjugate gradient with projection on the box constraints. The subprocedure MINX executes the direc tional minimization and is written in the form which can be applied to more general oroblems than those formulated in the model. The nongradient minimization method of Fibonacci is applied . The subprocedure FLAG calculates the value of the augmented Lagrangian (shifted penalty function) at a given point with given values of the shift and of the penalty parameter. FLAG utilizes the procedures of calculating the values of the state variables , of the function, of the constraints and of the right hand sides of the state equations . These subprocedures are calculated in the procedure MWP, outside the Solver . The Procedures M W P The set of procedures M \'I P reads in brief - models, values and derivatives . It consist of: the description of all so far investigated versions of the HOC models procedures which compute the values of the s t ate vec t or , of the objective functions,
Optimal Control Sectorial Model of the Polish EconoIllY of the equality and inequality constraints and of the constraints at time instant t-1, 'Jrocedures which compute the derivatives of the state equations in the control space, of the objective functions with respect to the controls and to the states, of the equality and inequality constraints at time tandt-1. All the data, necessary for solution of the model, including also the initial values, are called for in this set of procedures. The Output Procedure In result of a single solution of the MOC model, for given model parameters, structural matrices and desired (reference) paths we are obtainig values of the control variables Ut (the vectors Ct ' Vt , Et' Mt , Lt) values of the state variaoles x t la vector Kt and the scalar St) and of resulting va rlables : sectorial production Qt ·' net national income 0t' production andJinves~t imports, as well as the total imports Jm , for all investigated time instants. After the solution procedure is completed the OUtDUt subs e ts are created, which in turn pr~pare the data for the graphic system PLOTCALL. These subsets include: the solution results Ut' x t ' and based on then computed values of Gt · , 0t' Jm t , and some other values of selected lnput data, for instance lower bounds of investments, consumption and imports, and the desired paths of the economic categories which appear in the objective function. The above variables can be presented in the form of graphs on the computer monitor and printed. The graphs can be saved and the results can be compared later with the solutions of other versions of the model and with the input data. Exemplary, Tentative Canputational Results In the way of an example some preliminary results are presented in Fig. 3 - Fig. 6 . In Fig. 3 the investmens V . are presented t in the sectors of chemistryJand light ind~ try for the consecutive years 1985-1994 . Investments are changing fast over time. They are the most sensitive variables of the model. In Fig. 4 we present the value of fixed assets Kt · in the chemical industry dependent on ~~e value of investment Vt · . An increase of this value, (Vt2 ~ Vt1) ' do~s not allow to reach the desired traJec tory of fixed assets, which is extrapolated until 1995,V is higher than any investments 2 anticipated In the economic plan 1986-1990. Figure 5 presents the trajectory of the foreign debt S in American dollars. The desired trajectory is supposed to be con stant starting 1990. The presented trajectories of St show the dependence of the Polish economy on imports and credits. An increase of the investment (S21 is conp uted for V >V ) as well as an increase of con2 1 sumptlon (S12 is computed fOE hlgher des idered values of consumption C >C ) yields 2 1 higher values of the foreign dent than that of S11 ' computed for V1 and 1 . Lower bounds are assumed on the investments in the model. With no lower bounds, the inves~ ments path goes down with time as shown in Fig. 6 and might also assume zero values at some instants of time in selected sectors.
c
249
CONCLUSION The possibility of application of the optimal control model and of the multiobjective optimization to investigations of dy namic relations between the major economic variables of the national economy can be of great importance. The solutions of the model based on actual data can be utilized in the period which proceeds the construction of alternative economic development programs. The model itself, in spite of its deficiences, could be a useful and powerful tool in the decision making process. REFERENCES Bertsekas, O.P. (1976). 11ultiplier nethods: A survey , Automatica 12, 133-145. Bertsekas, O.P. (1982). Projection Newton method for optimization .problems with simple constraints. SIAII Journal on Control and Optimization. Vol. 20, No. 2, pp . 2 21 - 246 . Chmielarz, W., K. Cichocki, A.Stachurski , (1982), Experiments with the penalty scalarizinq function for non linear multiobjective optimization problem. Control and Cybernetics, Vol. 11, No . 1 2, pp . 57 7 2 . Cichocki , K. (198 5 ), Application of an optimal control model to investigations of economic policy in Poland, paper presented at the 5th World Econometric Congress, Boston, Aug. 1985. Cichocki, K., M. Lewandowska, A. Stachurski W. Wojciechowski, I. Woroniecka, Computer procedures for solution of the sectorial model of national economy, Yearly Report [in polish], Systems Research Institute, Polish Academy of Sciences, ZTS - 59 / CPBP 10.09, Warsaw 1988. Hesteness, M.R. (1969), Multiplier and gradient methods , Journal of Optimization Theory and Application, 4, 303-32~ Powell, H.J.O., (1969), A Method for nonlinear constraints in minimization problems. In R. Fletcher, (Ed.) Optimi zation, Academic Press, London , New York, Chapter 19 . Rockafellar, R.T., (1974), Augmented Lagrange multipliers function and duality in nonconvex programming, SIA.M Journal on Control, 12, 268-285. Wierzbicki A.P., (1979), The use of reference Objectives in multiobjective optimization - theoretical implications and practical experience , Technical Report, IIASA, Lasenburg, Austria, WP-79-66.
OPTIMAL CONTROL SECTORIAL MODEL OF THE POLISH ECONOMY WITH FLEXIBILITY IN CRITERIA AND CONSTRAINTS K. Cichocki S\'slems
Rl'Search
Inllilulf ,
Pulish Aradcmv of
SCifll(fS,
Nfll'l'lska 6, 01-447 Warsaw, Poland
Abstract. A nonlinear, sectorial model of the national economy is investigated. It is formulated as a multicriterion, o~timization problem with constraints in the form of difference equations, linear and nonlinear equalities and nonequalities. A scalarization function is a~plied to transform it to a discrete in time non-classical optimal control problem with box constraints on control variables. Flexibility is introduced in a~plying various constraints and various objective functions at the construction of alternative multicriteria optimal control (MOC) models. The objective in the model is to minimize the distance between desired paths (rates of growth of consum~tion, national income or other economic indicators) and the ones obtained as solutions of the model. Computer procedures are elaborated on the IBH PC. They include: main control program, descri~tion of several versions of the model with computation of values of constraints, objectives, state variables and their derivatives, input procedure, several versions of model solutions (dependent on the type of constraints) and the output procedure with graphics. Exemplary solutions are given for the period 1986-1995. Keywords. Modelling of economic systems, o~timal control, multiobjective and nonlinear optimization, com~uter sinulations, computer procedures. INTRODUCTION
periode or at least followed with a minimum distance.
The objective of the paper is two fold: first-to present various versions of a se~ torial model of a national economy, formulated as optimal control problems, and secondly - to describe a package of computer procedures on the IBM PC which describe and solve the above models. Flexibility is introduced in applying various constraints and various objective functions at the construction of the multicriterion optimal control (MOC) model of the economy. Flexibility is also incorporated in the solution procedurs of the nodel. It allows for solving multicriteria or single criterion problems with equality and (or) inequality constraints and (or) box constraints on the control variables. Additional linear difference equations can be included.
The objective of the model is to minimize, subject to constraints, a distance between the desired paths and the ones gene rated by the model. Given the desired paths (or a path) a single solution of the formulated optimal control problem yields the optimal values of control variables which satisfy the constraints and ensure that selected objectives are at the minimum distance from the desired objectives paths. The Eucleudian norm is introduced. The elaborated model belongs to a class of discrete-time optimal control problems with additional upper and lower bounds on the values of control variable s. The multicriterion problem formulated at the outset is transformed to a single criterion problem with an aid of a scalarizing (achievment) functions (Wierzbicki, 1979 and Chmielarz, Ci.c hocki, Stachurski 1982). We have selected various constraints and various objectives in order to construct versions of the MOC model of the national economy. So far eleven versions of the model have been programmed on the IBM PC and investigated.
The model is formulated as a nonlinear dynamic, multicriterion optimization problem with constraints in the form of difference (state) equations and linear and nonlinear equalities and inequalities. It is designed for assisting a deciSion maker in investigations of alternative economic development strategies and for checking the consistency of assumptions of economiC projects and plans. The assumptions concern rates of growth, over time, of basic economic variables which characterize the national economy. These variables include consumption, investment, evmployment, exports, imports and resulting from them - production, fixed assets, net national income and foreign debt. Hypothetical assumptions yield desired paths (rates of growth), for instance of consumption, or of national i~ come, which, in the view of decision I!'akers, should be achieved within a given time
The !lOC model can be applied in practice: firstly, for checking the consequences of a selection of various alternative desired "aths, secondly, for investigating "the inner consistency" of the model, i.e. for checking the feasibility of assum~tions of a simultaneous achievability of a number of desired ~aths, which represent desired growth rates. If the simultaneous achievment of the assumed growth rates is not possible-for instance requires high imports or (and) very high investments, then we conclude
245
246
K. Cichocki
that the assumptions of a development plan are not consistent (it can be true in both cases, of feasibility and infeasibility of solutions). A hypothetical way of utilisation of the MOC model in a decisiona-aiding process, for instance by a decisiona maker in investigating and elaborating an economic plan, is presented in Fig. 1. The ?lanner takes into account the world market situation, home economic conditions (resources, technology and technological progress) and formulates an initial development scenario, based on his knowledge of the capacities of the economy and of expectations of the society. The "initial scenario" is verified in result of its inner inconsistency or due to investigations of a hypothetical development scenario. In the process of confrontations of the planners goals and intentions, expressed by the desired paths of growth and the results yielded by solutions of the model, the planner can change the values of the desired paths or he can choose new goals of the economic developnent. For instance he can either try to satisfy a given consumption level or to diminish the foreign debt, or to attemp to do both at once. Another objective can be the maximization of employment in the economy. Computer procedures in FORTRAN on the mM B:::: have been elaborated. Their purpose is to describe the set of the MOC models of the national economy, enable to solve the model, conveniently change the input data and present the model solutions. The set of procedures c~nsist of: the main control program (MAIN), input procedure (INPUT), procedure of the models description wi~h supporting computational procedures for instance of computing the values of state variables, constraints and of objective (MWP) , the solution method (SOLVER) and the output pr~ cedure (OUTPUT) with suitable graphiCS. THE MOC MODEL OF THE NATIONAL ECON0I1Y Description of the Model The multicriterion optimal control model, with slight simplifications, can be written in the form min f(x(u), u) where f=(f , ... ,f , ... ,fn) , n>l (1) 1 i u and f : RlT x RsT -> Rl , i = 1, ... ,n i
ST U E Un C R . The function f transforms the feasible set into the set of achievable pOints Q since, given u the relaO tion x(u) is uniquely defined by equations (2), (3), QO=f (U D) . The multicriterion problem (1) - (6) is transformed to a single-criterion one with the help of a scalarizing function method (Wierzbicki 1979). Minimization of the scalarizing function J(u,f) substitutes for minimization of n criteria of the form (1). The function J(u,f) is of the form n
2
_
J(u,f) = p1: Gi[max(O,fi-f (x(u) ,u))] i i=l n
- 1: 0 i[fi(x(u) ,u» i=l
(7)
- fi]2
where f. are desired paths (reference-paths)~ which are determined exogeneously, 0 . are weighting coefficients and p >l is a~penatly parameter. _ n Changing the desired path f E R one can obtain any path f EQ . If f is_not achievO able the minimization of J(u,f) corresponds to the minimization of n _ 2 (p-1) 1: c .(f.-f.(x(u),u» i =l ~ ~ ~ while when it is achievable, the minimization J(u,f) corresponds to the maximization n
of 1: (f.(x(u),u) _f)2. i=l ~ Economic Interpretation of the Hodel In the MOC model we define the following variables: control variables, which include the vector of consum?tion C = (C , ... ,C ) , the investment vector t1 tm t Vt = (V t1 ,··· ,V tm ) , exports E t = (E t1 ,··· ,Etm) , ~mporEs for consumpt~on purposes Ht = (/\1' ... ,11 tm ) and employment L = (L t1 , ... ,Lt)' where m is the number ot sectors in ~he economy and t=O,l , ... ,T-l; T denotes the length of the investigated period. state variables which are functions of the control variables include the vector of fixed productive assets Kt = (K t1 , ... ,K ) tm and the scalar value of the fore~gn debt St' defined for the whole economy, where t=O,l , ... ,T.
gj(Xt , Ut) ::; 0
j
1, ... ,p
(4)
gk(ut ) = 0
k
1I
,r
(5)
The linear difference equations describe the behaviour of the state variables over time. There are m equations of ty?e (2), which describe accumulation of fixed assets. At time t+l it deoends on the value of fixed assets at time t decreased by a part of it which is depreciated, and on distributed over time investments. Part of the fixed assets 6 determined over the time (O,T) is due to investments committed prior to time t=O.
t
0,1, ...
(6)
Kt +1 ,j=(1-d jj ) Kt + T: /T J Vt- T,J + 6 tj
subject to constraints N ht (u _ ) , x t +1 = q,x t + 1: T t T T=O x x 0
t=O, 1 ... ,T-l
(3)
0
~t ~ Ut
$
(2)
•••
t
Ut'
Ut > 0;
I
T-l
where n is a number of criteria, ht and gk are l~near functions, g. is a nonlinear function, q, is a constarlt matrix of the dimension 1 x 1; u~ is a vector of control variables, Ut E R , and x is a vector of t state variables,
The controls u belong to a feasible set U , determined by constraints (2) - (6); D
t
(8)
with given initial value j
=
1, ... ,m
( 9)
The values of ~ t. are time dependent coefficients of invJ~tments delays. The accumulation of debt St+l is described by a single difference equat~on which explicitly depends only on past credits (value of the debt at time t), imports and exports.
Optimal COlltrol Sectorial Model of the Polish EcollolllY
m 5t+1=(1+rt)5t+j:1 [GtQtj + (HtVt)j + Mtj - Etjl with given 50 = 50
247 (10)
The value of r t represents the interest where Tl is the liquidat10n time of an agrates on crediEs, while Q , is the value gregate investment, for instance of cont of production of the j-th ~ector derived structions in the chemical industry. If we from the balance equation do not want real decapitalization of fixed m m Qtj - i:1 atjiQti + gtjjQtj + Mtj-Etj-aj(0tj-0t_1 ,j)=i:1 (btji-btji) Vti + Cti ; j=1 , ... ,m; t=O, ... ,T-1 (11) where a" are elements of the intermediate material1coefficients (Leontief) matrix, g" are elements of a diagonal matrix G ofJthe average propensity to import for oroduction purposes, b" are the total and t" imported capital irt~estment coeffients irt1the j-th sector. Ht is a diagonal matrix of the total imEorted investment coefficients h = b . The coefficients a <1, j ji jj
i
determine a part of a production increase which defines the reserves. The production capacities are approximated by a nonlinear constant elasticity of substitution (CES) production function. It is assumed that the production distributed to sectors of the economy can not exceed production capacities (12) where mt' is the coefficient of utilization of Jproduction capacities, O::illltj ;S 1. The consumption in all sectors is greater than a given consumption subsistence leveL The other control variables are assumed to be either nonnegative or lower bounds on their values are assumed. Usually in all but one version of the model full employment in the economy is assumed. The summation of employment over sectors yields a time - variant employment in the economy.
assets we can require as in the 3rd version of the model that (16) The constraints (15), (16) do not show up in the versions 1 and 2 of the model, in which it is required in the objective function that the fixed assets follow given exogeneously desired trajectory. The 11th version of the model allows for unemployment. Its objective is to maximize employment while in another, not included version of the model we can maximize simultaneously employment and consumption. The arrows in Table 1 indicate selected ways of carrying out comparisons between consequent versions of the models. Firstly one can investigate implications of the selection of various goals, which are to be achieved by the economy, and consequences of introducing or relaxing the constraints related to the level of fixed assets, new credits, foreign debt, the national income etc. Secondly, one can select various desired growth rates of the consumption, national income, fixed assets, foreign debt and employment and investigate the achievability of thes e desired paths. In case of selection of several desired paths at a time we can check if a simultaneous achievement of these paths of desired growth rates is possible. If the desired growth rat e s which can represent assumptions of a given economic d e v e lopment project (for instance of the economic plan 1985-1990), can not be achieved simultaneously, then we conclude that the assumptions of the plan are not consistent.
In Table 1 we present eleven versions of the MOC model which have been programmed and investigated. They differ, one from another, in the form of the objective function J(.) (the number of objectives selected) and in the number and (or) form of DESCRIPTI ON OF COMPUTER PROCEDURES the constraints. For instance the 6th verOF THE MODEL sion of the model assumes the objective function of the form J(D,D)in which we are The Input Procedure interested only in the growth path of the net national income Dt , while the 7th verThe procedure allows for introduction of sion allows for invesEigation of the growth suitably prepared sets of data which are rate of both the consumption and the nationecessary to solve the MOC-model. The set nal income paths. In this case the objecof data is divided into more than twenty tive assumes the form T-l m T-1 _ 2 T-l m _ 2 T-l _ 2 _ 2 J(C,D,C,D)= p { 2: l: ,[max (0 'CtJ--{;t ,) 1 + l: OD [max(O,Dt-Dt ) 1 } - { l: ,2: 0c,(Ct'--{;t') + l:0D(Dt-Dt ) } =1 cJ J j J=l J J J t=O t=O t =O t=O p>l (13)
°
where the nett national income m m Dt = l: (Qt' - , l: atjiQti' t=O,l, ... ,T-l j=l J 1=1 i;
(14)
In both versions, 6th and 7th we assume that the yearly increase of the foreign debt (including new credits) should not be greater than 25 per-cent of the total exports at a given year. Additionally, a lower bound on fixed assets is assumed. It results from a linear rate of depreciation (liquidation) of capital, provided no new investments are committed in the investigated period, i.e. o ' Kt , ~_K to ,-K to ,t-t (15) -m---r- + 6 , , J=l, ... ,m tJ J ,J J ~lj-LO
subsets, each including several data units. A single unit constitutes for instance a matrix of dimension 15xl0 whose elements are e.g. desired values of consumption over a ten years period for fifteen sectors of the economy. Another data unit can be a material coefficients matrix of dimension 1 5xl 5. The introduction of the data is organized with a cooperation of the user who selects a data unit in each subset. The size of the matrices and (or) its elements can be changed. New data units can be established. Changing the data of a given data unit is done outside the Input procedure. This procedure will be more user friendly.
248
K. Cichocki
TABLE 1
The comparison of various !10C models
2
3
4
5
J(C,K,S) J(C,K,S,D) J(C,S,D) J(C,S,D) ,J(D)
6
J(D )
8
7
J(C,D)
J(C,D)
9
J(C)
10
J(C)
11 J(L)
K
Kt~tO
K
T6t-t KtO .......................... Kt ->Kto - ~ T -t Kto I 0 I 0
C
S
D
L
l:Ltj = It .......................................................................... l:Ltj= l t l:Ltj ;;lt
/. The Solution Procedure of the Model The so l ution method is elaborated and programmed in the form which enables the solution of a discrete in time, optimal control problems with constraints on the control and the state variables also of the type described by the MOC model. The method is structured in a flexible way . Dependent on a value of a special parameter of ITRYB it solves either a problem with box constraints on the control variab l es only, or a problem with additional inequality constraints (or inequalities and equalities) . Additional possibilities include linear state equations and equality and (or) inequality constraints. The nonlinear, discrete optimal control problem (2) - (7) is solved with the help of the augmented Lagragian method (shifted penalty function method) - (Bertsekas, 1976; Hesteness 1969; Powell 1969; Rockafellar, 1974). The Lagrangian is minimized in the control space. The linear state equations (2) with the initial conditions (3) enable exact determination of the state variables x , given controls u. Thus, the problem can be reduced to the control space, reducing also the dimensionality of the problem. The reduced gradient of the augmented Lagrange functional can be computed and the ajoint variables determined since the linear state equations (2) are explicit linear equality constraints with value s on the state space_ Thus, nonzero multipliers for those con strainte exist . For given " penalty parameters " we are mini mizing with respect to controls u the augmented Lagrange functional subject to the bounds on the control variables (6). The applied minimization bases on the Newton projection method for minimization of twice differentiable functionals subject to box constraints and is taken from Bertsekas (1982). However, in the subspace of free variables we do not use the Newton direc tion . Either a decent direction is used when the active constraints change or a conjugat gradient direction otherwise. The directional minimization util iz es the non-gradient method of Fibonacci. It results from the fact that the functional dependence
of the value of the minimized function and the length of the directional step is not necessarily differentiable , because of the projection. Thus, dependent on the value of the ITRYB parameter we are applying either a method of the conjugate gradient with projection on the box constraints or the augmented Lagrange method with the r educed gradient or the augmented Lagrangian method only . The solution procedure consists of several subprocedures which are shown in Fig.2. The subprocedure I1UST minimizes the Lagrange functional in the control space, utilizing the augmented Lagrangian method and the conjugate method with projection on the box constraints. The reduction of the problem to the control space is initiated or executed in the subprocedure FLAG or in GRAD. The subprocedure GRAD calculates the value of the reduced gradient of the functional in the control space (computes the adjoint variables) . The subprocedure PROJG determines the working set of active constraints in the k -th iteration of the conjugate gradient with projection on the box constraints. The subprocedure MINX executes the direc tional minimization and is written in the form which can be applied to more general oroblems than those formulated in the model. The nongradient minimization method of Fibonacci is applied . The subprocedure FLAG calculates the value of the augmented Lagrangian (shifted penalty function) at a given point with given values of the shift and of the penalty parameter. FLAG utilizes the procedures of calculating the values of the state variables , of the function, of the constraints and of the right hand sides of the state equations . These subprocedures are calculated in the procedure MWP, outside the Solver . The Procedures M W P The set of procedures M \'I P reads in brief - models, values and derivatives . It consist of: the description of all so far investigated versions of the HOC models procedures which compute the values of the s t ate vec t or , of the objective functions,
Optimal Control Sectorial Model of the Polish EconoIllY of the equality and inequality constraints and of the constraints at time instant t-1, 'Jrocedures which compute the derivatives of the state equations in the control space, of the objective functions with respect to the controls and to the states, of the equality and inequality constraints at time tandt-1. All the data, necessary for solution of the model, including also the initial values, are called for in this set of procedures. The Output Procedure In result of a single solution of the MOC model, for given model parameters, structural matrices and desired (reference) paths we are obtainig values of the control variables Ut (the vectors Ct ' Vt , Et' Mt , Lt) values of the state variaoles x t la vector Kt and the scalar St) and of resulting va rlables : sectorial production Qt ·' net national income 0t' production andJinves~t imports, as well as the total imports Jm , for all investigated time instants. After the solution procedure is completed the OUtDUt subs e ts are created, which in turn pr~pare the data for the graphic system PLOTCALL. These subsets include: the solution results Ut' x t ' and based on then computed values of Gt · , 0t' Jm t , and some other values of selected lnput data, for instance lower bounds of investments, consumption and imports, and the desired paths of the economic categories which appear in the objective function. The above variables can be presented in the form of graphs on the computer monitor and printed. The graphs can be saved and the results can be compared later with the solutions of other versions of the model and with the input data. Exemplary, Tentative Canputational Results In the way of an example some preliminary results are presented in Fig. 3 - Fig. 6 . In Fig. 3 the investmens V . are presented t in the sectors of chemistryJand light ind~ try for the consecutive years 1985-1994 . Investments are changing fast over time. They are the most sensitive variables of the model. In Fig. 4 we present the value of fixed assets Kt · in the chemical industry dependent on ~~e value of investment Vt · . An increase of this value, (Vt2 ~ Vt1) ' do~s not allow to reach the desired traJec tory of fixed assets, which is extrapolated until 1995,V is higher than any investments 2 anticipated In the economic plan 1986-1990. Figure 5 presents the trajectory of the foreign debt S in American dollars. The desired trajectory is supposed to be con stant starting 1990. The presented trajectories of St show the dependence of the Polish economy on imports and credits. An increase of the investment (S21 is conp uted for V >V ) as well as an increase of con2 1 sumptlon (S12 is computed fOE hlgher des idered values of consumption C >C ) yields 2 1 higher values of the foreign dent than that of S11 ' computed for V1 and 1 . Lower bounds are assumed on the investments in the model. With no lower bounds, the inves~ ments path goes down with time as shown in Fig. 6 and might also assume zero values at some instants of time in selected sectors.
c
249
CONCLUSION The possibility of application of the optimal control model and of the multiobjective optimization to investigations of dy namic relations between the major economic variables of the national economy can be of great importance. The solutions of the model based on actual data can be utilized in the period which proceeds the construction of alternative economic development programs. The model itself, in spite of its deficiences, could be a useful and powerful tool in the decision making process. REFERENCES Bertsekas, O.P. (1976). 11ultiplier nethods: A survey , Automatica 12, 133-145. Bertsekas, O.P. (1982). Projection Newton method for optimization .problems with simple constraints. SIAII Journal on Control and Optimization. Vol. 20, No. 2, pp . 2 21 - 246 . Chmielarz, W., K. Cichocki, A.Stachurski , (1982), Experiments with the penalty scalarizinq function for non linear multiobjective optimization problem. Control and Cybernetics, Vol. 11, No . 1 2, pp . 57 7 2 . Cichocki , K. (198 5 ), Application of an optimal control model to investigations of economic policy in Poland, paper presented at the 5th World Econometric Congress, Boston, Aug. 1985. Cichocki, K., M. Lewandowska, A. Stachurski W. Wojciechowski, I. Woroniecka, Computer procedures for solution of the sectorial model of national economy, Yearly Report [in polish], Systems Research Institute, Polish Academy of Sciences, ZTS - 59 / CPBP 10.09, Warsaw 1988. Hesteness, M.R. (1969), Multiplier and gradient methods , Journal of Optimization Theory and Application, 4, 303-32~ Powell, H.J.O., (1969), A Method for nonlinear constraints in minimization problems. In R. Fletcher, (Ed.) Optimi zation, Academic Press, London , New York, Chapter 19 . Rockafellar, R.T., (1974), Augmented Lagrange multipliers function and duality in nonconvex programming, SIA.M Journal on Control, 12, 268-285. Wierzbicki A.P., (1979), The use of reference Objectives in multiobjective optimization - theoretical implications and practical experience , Technical Report, IIASA, Lasenburg, Austria, WP-79-66.