Two-level national-regional planning and mathematical programming decomposition applied to spatial price equilibrium models†

Socio-Econ. Plan. Sri.. Vol. 12. pp. 251-266 0 Pergamon Press Ltd.. 1978. Printed in Great Britain

AND

TWO-LEVEL NATIONAL-REGIONAL PLANNING MATHEMATICAL PROGRAMMING DECOMPOSITION APPLIED TO SPATIAL PRICE EQUILIBRIUM MODELSt H. PACCA LOUREIRO LUNAS Universidade

Federal de Minas Gerais, Belo Horizonte, Brazil

(Received 12 December 1977; received for publication 30 March 1978) Abstract-This paper is concerned with computation strategies related with the quantitative analysis of only a sector of a global economy (e.g. agriculture or energy). Under a ceteris paribus condition on the environment of the sector, and within the hypothetical context of the neoclassical economic theory, it is well known a type of partial equilibrium model that can be cast mathematically into an optimization framework. We attempt here to a typical specification which considers spatially separated markets, that has been called spatial price equilibrium model and that has contained within it many classical transportation problems (one for each commodity). The model is specially suitable for mathematical programming decomposition, resulting regional sybsystems whose coupling variables are the transportation flows. We explore here this structure, discussing two decomposition algorithms with economic interpretations that suggest decentralized procedures for planning. The first-of the price coordination type-is a variant of the Dantzig-Wolfe’s principle which is expected to have a low number of cycles of information flow between the master level and the sub-problems (at each cycle it is calculated a series of regional production-consumption responses to alternative prices generated by transhipment problems at the central planning level). The second algorithm specializes the Geoffrion’s projection/feasible directions technique, so interpreting the problem solution within the context of a net-output target coordination.

1. INTRODUCTION

Study of resource allocation in disagregative decentralized systems has been the main concern of the neoclassical school in economic theory. Nowadays, perhaps the most fully articulated mathematical formulation of this theme is the Walrasian model of competitive equilibrium, as it has been studied appealing to convex structures (e.g. see [I]). The existence of a solution for a general model now is proved, and also a numerical method to find a general competitive economic equilibrium is known, based on a procedure for the approximation of a fixed point of a continuous mapping [2]. As opposed to this computation strategy, mathematical programming techniques are not able to solve such a general model which in fact involves the interaction among many maximization problems, separately pursued by different economic agents, rather than being a problem of a single maximization of an indicator of social preference. But if one rules out some more stringent conditions of the general equilibrium model, and if one is interested in the quantitative analysis of only a sector of the global economy (for instance, the agricultural or energy sector), then it is possible to define a type of partial equilibrium model that can be cast mathematically into an optimization framework. In the latter case, model estimation is yet a difficult but a workable task, and large-scale versions can be solved by efficient techniques of mathematical programming, so enabling operational research applications at the level of governmental analysis and policy evaluation; we discuss here some of these sectorial equilibrium models. tThis work is partially sponsored by “Conselho National do Desenvolvimento Cientifico e Tecnol6gico”. SPresentaddress: Laboratoire d’Automatique et d’Analyse des Systtmes du C.N.R.S., Toulouse. France.

Samuelson[3] has originally proposed an artificial criterion (so called “net-social pay-off” and in similar variants being named “consumers surplus” or “net quasiwelfare function”) that permits a mathematical programming formulation to solve a partial competitive equilibrium. The clue consists of finding a mathematical program whose optimality conditions exactly meet the competitive conditions which one is looking for. Building on this idea, he handled the onecommodity problem of searching for spatial prices and flows involved among spatially separated markets, with supply and demand functions being given at each place; since then it is known that this problem of interconnected markets contains within it the Hitchcock-Koopmans’ minimum transportation cost problem [4]. An extensive survey on further development of this field is the book by Takayama and Judge[S]; by using linear price dependent supply and demand functions they have suggested the computation of the competitive (or even monopolist) solution by quadratic programming. Together with linear activity analysis models of production, which serve to implicitly generate supply functions, the optimisation approach has been used on practical multicommodity models of sectorial economic equilibrium: some to be remarked make analysis of agricultural sectors, and are, either spaceless versions like that of France[6] and Mexico [7], or are spatial models as that of U.S.A. [8,9]; also the same mathematical structure has provided a foundation stone for applied work on energy policy studies[lO, 111. Apart the use of simplex methods for quadratic programming [5,9] calculations for such problems have also appealed to conventional software of linear programming, either by means of explicit gridlinearization techniques [6,7], or by using adaptative demand iterative procedures (a recent insight in this approach is the appendix of [ 111). 251

252

H. PACCALOUREIRO LUNA

This paper is concerned with the study of large-scale mathematical programming techniques [4, 12,131 applied to sector-wide national economic models of spatial equilibrium. To some extent, it seems that the use of decomposition on such models has been neglected, although one may hope for two practical advantages explored here: (1) Alternative decomposition programs can be selected in order to have direct economic interpretations related with administrative schemes that integrate national and regional planning. Thus the economic equilibrium computation strategy can be performed within the context of decentralized procedures for planning; and for this purpose, possibly one might put into operation algorithm versions with not large number of necessary cycles of information flow between the master problem (central planning agency) and the sub-problems (regional offices). (2) The actual state of computational experience suggests that to solve some large-scale problems global methods would usually be quicker and simpler than decomposition techniques. Even though it may be expensive to solve other extremely large-scale problems, in certain cases it is inevitable the use of decomposition methods, either because of limited computer capacity, or in view of reliability (questions of recovering incorrect data, or of diminishing numerical errors may eventually be justifications). Spatial price equilibrium models easily fit into the class of very big size problems; for instance, as noted with caution by Hall et al.[9], their quadratic programming experiment encountered both numeric difficulties and excessive computer time, despite several model simplifications forced by a limited program capacity. As follows, perhaps dimension would require decomposition strategies if the problem is to be solved within available computational resources. Moreover, though many experiments are yet to be made to confirm computational efficiency, a spatial price equilibrium model has a structure specially suitable for mathematical programming decomposition. In fact, if one holds temporarily fixed every regional excess demandTa vector of differences between regional levels’ of consumption and production for each class of goodsithen the problem is divided into two highly separable parts: on the one hand, each regional economic subsystem-by itself having the two separated sectors of production and consumption-is isolated from the other regions; on the other hand, a series of classical transportation problems (one for each commodity) corresponds to an efficient coupling of all the regional economic sub-systems. As a matter of fact, in view of the hierarchy of decision levels, the socio-economic organization is arranged on structures of multilevel economic planning. Among a rather vast literature on the subject, a critical overview has been made by Kornai[l4], and, e.g. Malinvaud (Section III of [15]) revises procedures based on progressive price adjustments as suggested by the Walrasian tatonnement process. Such procedures have historical antecedents as proposed schemes to determine the economic equilibrium, and from the hierarchical control point of view they may be specified by the two following conceptual levels: -At a superior level, the role of coordination is played by a market (in a competitive economy) or by a central agency (in the Taylor-Lange’s model of a socialist economy). In any case, at each stage the coordinator revises given prices, adjusting upward the price of those

commodities whose excess demand is positive and decreasing the price of the commodities for which the opposite holds true. -At an inferior level, a number of consumers and a set of producers (firms) receive a proposed price for every class of goods. The typical consumer will calculate his demand for each of the commodities by maximizing his utility subject to a budget constraint. And each firm selects its proposal for net production in such a may that it is technically feasible and maximizes the firm’s profit as calculated with the given prices. As a result of these consumption and production responses, the coordinator will know the excess demand corresponding to the prices informed in the previous cycle. The above outlined historical procedure normally has not shown good computational efficiency; also, convergence to the economic equilibrium can not be proved without stringent assumptions (for brief discussion and references, see [2]). But it has provided a background for much further work on possible structures of multilevel economic control. Actually, not only many other price coordination schemes are known (perhaps the most important corresponding to economic interpretations of the Dantzig-Wolfe’s decomposition method; e.g. see Section V of [15]), but also resource-directive strategies-with central allocation of input quotas or output targets-are well introduced (see Geoffrion[ 121). In this paper, Section 2 presents formulations of spatial price equilibrium models-including a practical linear-quadratic version and a nonlinear extension with convexity requirements-giving their optimality (or economic equilibrium) conditions; Section 3 attempts to use a variant of Dantzig-Wolfe’s decomposition to solve the spatial model within a price indicative planning procedure; and Section 4 specializes the Geoffrion’s projection/feasible directions technique (as proposed in Section IV of [16]), so interpreting the problem solution within the context of a net-output target coordination. In both Sections 3 and 4, in order to overcome difficulties of treating interconnected market places, we make an extensive appealing to properties of the classical transportation problem. In as much as the distribution agents competitive behavior may be reflected by a transportation problem, in fact this problem constitutes more and more an element of linkage among spatially separated markets. And for the sake of using transportation problems as basic tools to analyse links among regional economic sub-systems, a model partition exists such that the interregional flows are associated with the primal variables of transportation problems whose dual variables are easily interpreted as regional prices. Accordingly, in some extent and from another viewpoint we are here concerned with solution methods for a sort of “extended multicommodity transportation problem” whose own supply and demand quantities are generated as responses to correspondent set of node price values. As opposed to other extensions of the transportation problem which include integer O-l decision variables (e.g. see [17]), the extension treated here makes no attempt to structural changes on the production and distribution network, so having only continuous variables. 2. SPATIAL ECONOMIC MODELS 2.1 Quadratic programming problem Operational versions of spatial price equilibrium models have been obtained with quadratic programming formulations[8,9,5]. In fact, such large scale programs

Planning and mathematical

programming

are almost linear, since they are usually composed by the following three parts: (1) a quadratic partition in the objective function that implicitly represents consumers’ demands, (2) a linear activity analysis model of production for each region, and (3) a linear distribution component which enables material balance among the regions and the international market. A simplified typical specification can be made using the notation that follows [a capital letter denotes a matrix; a small letter overscripted indicates a vector but if it is also subscripted then it indicates a component of the vector]: Constants (of the linear parts): r, 7 index for regions (r, f = 1,2,. . . , R), n index for final (or desired by consumers) com-

modities (n = 1,2,. . . , N), index for primary commodities (resources), index for production activities. a line vector of activity costs in region r, where, each component c; represents purchase of fixed price inputs from other sectors of the economy; these inputs are necessary to operate activity j at a unitary level and are not included among the restricted resources; b’ a column vector of resource availability levels at region r; a component b,’ is the quantity of primary commodity m disponible at region r and that can not be transferred to other regions (or, assumed to be an immobile resource); a typical example for the agricultural sector is land. A’ a matrix corresponding to the output partition of the technology matrix of region r; an element aLi gives the output of desired commodity n when activity j is operated at level one: B’ a matrix associated with the input partition of the technology matrix of region r; a coefficient b,$ denotes the required input of resource m in order to operate activity j at an unitary level. d a line vector of transportation costs, eventually including taxes and subsidies; a component da(or d.‘“) indicates the cost to transfer a unit of product n from region r to P (or, from r to exportation); for simplification, suppose d.” = 0 for every r and all n ; P0 a line vector of the international market prices; the index n of a component pnOrefers to class of goods n. m i c’

Variables : X’

4’

f P’

U’

a column vector where an element xi’ indicates the level of production activity j operated at region r(x’,r=l,...,Rcomposex); a column vector of final commodity quantities which are desired by consumers in region r, a subscript n referring to the correspondent commodity (q is partitioned on q’, r = 1,. . . ,R); a column vector of transportation activity levels; for instance, f.“’ denotes the flow of commodity n imported and going into region r; a column vector of desired commodities prices for consumers of region r; the price of product n is given by the component p.’ (p is a R x N-elements vector composed by all p’, r = 1, . . . , R); a column vector of imputed prices for the primary commodities in region r; u,’ refers to price of resource m(u’. ~, r = 1.. . . . R are sub-vectors of a)., I

I

253

to spatial price equilibrium models

Demand functions :

For every region, it is supposed known a linear price dependent demand function for desired comodities, i.e. q’=q”+Q’p’,

(1)’

r=l,..,R,

where q” is a N-component positive vector, and Q’ is a square matrix assumed to be symmetric and negative definite. [Here and after, the prime on the equation number (1)’ means that this equation is the simplified (linear) version of another more general (nonlinear) relation that will be labelled with the same number but without prime.] The later hypothesis assures the existence of the inverse matrix Q’-‘, so there is a one-to-one correspondence between q’ and p’ and the following inverse function can be defined: p’=p”tP’q’,

r=l,...,R

p”’ = - Q’-‘qc’ and P’ = Q’-‘, implying that P’ is also a symmetric and negative definite matrix. Noting by the vector q’ the transpose of q, now may be defined the following quadratic programming

where

problem:

(3) Minimize

s,XHLfro ’

p”+;P’q’)+c’x’]+df + p0 (xf”’

r

(4)

(5)

q’-A’x’_

&(fa'-f'v0 B’x’sb’

- xfrO)

r

. I

subject to

r=l 3. . . ,R

I

where a represents any node of the distribution network ((Ymay be either the node P of a region Pf r or the node o of the international commerce terminal) and L(r) is the set of nodes which are linked with r. For a compact formulation, define T as a multicommodity incidence matrix relating interregional flows and levels of regional commerce with other countries. A column representing a possible flow of commodity n from r to f has a coefficient t 1 in the row associated with region r and product n, a coefficient - 1 in the row corresponding to region P and commodity n, and zero elsewhere; a column indicating commodity n importation into region r has a coefficient - 1 in the row associated with region r and class of goods n, and zero on every other row; a column denoting commodity n exportation out of region r has a coefficient t 1 in the row corresponding to region r and product n, and zero elsewhere. Also defining A and B block-diagonal matrix respectively composed by the output partition A’ and the input partition B’ of every regional. technology matrix, as illustrated on Table 1, a compact specification is: (3) Minimize - q’p’ + 1 q’Pq) + cx + Sf subject to q.xZ’clf~O (4) (5)

q-Ax+Tf>O Bxsb

where the auxiliary vectors c,p’, b are respectively partitioned on the regional vectors c’, p*’ and b’ defined above; and S is a line vector representing costs for every

H. PACCALOUREIROLUNA

254 Table

1. Tableau

related with the linear part of the spatial equilibrium model

(14)’ p”[q’-A’?r’-.~~))Cf.‘-f-)]=O

r= 1 , .., R 1

(15)’ u”(b’ - B’x’) = 0

I

The optimality conditions outlined above correspond to a competitive equilibrium in the economic sector that is studied (e.g. let’s fix idea on the agricultural sector, including the activities of production, distribution and consumption). In fact, eqn (2)’ fulfils the estimated econometric dependence between the regional vectors of 8R Q bR prices and consumers demands. The restrictions (4)’ c’ c’ CR % assure no excess demand for every region, and its slack complementarity (14)’ permits a positive regional price only if there is no excess supply. The constraints importation (5)’ maintain regional activites within its production possibility set, and (15)’ enables a positive price only for a scarce resource. The conditions (6)’ state Production that no production activity can make a profit in equilibrium because capital would otherwise move into the activity; note that each element of the vectorial inFig. 1. Sketch of the distribution structure embedded in the spatial equality (6)’ asserts that the total cost to operate an activity (the left hand side, representing the addition of equilibriummodel. the cost associated with nonrestricted inputs summed with the value of restricted resources) is not smaller than arc in the network, given the transportation cost for the benefit resulted from the operation of the activity interregional flows, the transport cos: summed with the (the right hand side, corresponding to the output value of international price for importation flows and the trans- a unitary operation level). The slack complementarity port cost discounted by the international price for eqns (10)’ affirm that only activities which breakeven can exportation flows. P is a block-diagpnal matrix whose be operated at a positive level in a competitive equilisubmatrix are the above defined P’, r = 1,. . . , R. brium. And finally, the constraints (7), (8) and (9), together with its correspondent slack complementarity eqns (11) (12) and (13), stipulate conditions for equilibrium in the competitive behavior of the distribution 2.2 Optimality conditions and economic equilibrium Problem (3)‘-(5) searches for a minimization of a agents. There is no arc where a transportation activity convex objective function subject to linear constraints, could give a positive profit, and it is possible a positive so the Kuhn-Tucker conditions are necessary and flow only through an arc whose difference between sufficient for a global minimum. Or, at an optimal solu- destination and origin node prices is exactly equal to its tion, the primal variables q (unrestricted) and x,f (both transportation cost. Now some general remarks on this class of models non-negative) satisfy inequalities (4)’ and (5)‘; yet, there are non-negative dual variables p and u such that the together with some particular features of the present following relations are also verified and assure optimality formulation deserve mention: -At first, this type of model must be applied under a together with (4)’ and (5)‘. ceteris paribus hypothesis, by supposing that everything p’ = p#’ + P’q’ else in the environment of the studied sector remains (2)’ constant, and yet this sector is relatively small so that it r=l ,...I R, cr + nr’Br 2 p”A’ has no sensible influence in the overall equilibrium of the (6) economy. For instance, no matter what might be the endogenous agricultural revenue (or the farmer’s in(7) p”-p”sd”, for every set (r,P) come), the use of eqn (2)’ presupposes a given regional = (r, P)J(r, F),, is an arc associated consumer’s income. Also the use of constant costs for production and distribution activities (the vectors c and with flow of product I d) presumes a fixed price for every input necessary to ( n from region r to f operate these activities, no matter what agricultural (8) p” 2 p” - d”, for the set (r, o) of exportation arcs. decisions might really influence prices of products from other sectors of the economy. (9) pr' s p“ + d”‘, for the set (0, r) of importation arcs. -Another point is the artificial character of the objective function, which has not been proposed as a direct The relationship among the gradient vectors of the obmeasure of social preference; instead, it is simply jective function and of the c
Planning and mathematical

0

4”

programming

4;

Fig. 2. Relation between price and quantity of commodity n in region r, assuming a demand function with no interdependence among products.

demand are zero for every region), then the quadratic part of the criterion (3)’ is a measure of consumers’ welfare and it may be illustrated by Fig. 2. In fact, the shadowed area of Fig. 2 depicts the value of the parcel corresponding to commodity n and region r in the quadratic part of (3)‘. If equilibrium occurs at point E, the welfare accruing from the consumption of commodity n in region r would result from the addition of two portions: the inferior rectangular area gives the economic value or revenue of the product (price X quantity), and, the superior triangular area indicates the consumers’ surplus or a measure of how much more the consumers receive than they pay for. In this case (3) may be interpreted as the criterion of maximizing the net-benefit value”: following “nationalsectorial consumers’ welfare plus exportation earnings minus production and distribution costs and minus importation expenses, all these partial values referring to products of the sector under analysis. -Note also that our model formulation has no balance of payments restriction. For example, it is expected that if it is applied to the agricultural sector of a rural oriented economy it will result sectorial earnings on foreign exchange, as well as if it is applied to the energy sector of an oil dependent country it will indicate a deficit on the balance of payments. Therefore, the model is consistent with an international commerce without quota imposition. Moreover, it permits the establishment of lower and upper bounds for every price, as it arises as a consequence of conditions (8) and (9). The upper bounds will always be positive since one has defined positive importation prices at the international com-the maximal price of commodity merce terminal; p LmaX n in region r-can be computed by adding the international price of n(p,“) and the cost (CL”‘) to transfer product n from the international market to region r. The lower bounds--p Lmm being the minimal price for commodity n in region r-could be positive if one assumes that each region r can transport any of its product n to the international market o affording a cost d,” inferior to a positive exportation price pnO.Yet is expected that in empirical applications the demand eqn (1)’ would bear positive q’ for every p’ contained within the stipulated bounds, so restricting prices and quantities within ranges of economic sense, as illustrated in Fig. 2, where A and Z represents extreme cases of equilibrium points; (remark that model (3)‘~(5)’ takes q as an unrrestricted vector in

to spatial price equilibrium models

255

order to meet the equality constraints (2)‘). The role of price bounds will be important on our algorithmic developments. Note that if it results from the international commerce one must presuppose a small country which acts as a price-taker and has no influence on the international market prices; but the same structure of price bounds could be obtained within the context of price support policies and governmental stocks for price stabilization. -A third point concerns requirements upon the matrix Q’ of the demand function (1)’ and on its estimation. According to previous works (see [9,18]), a complete economic interpretation of the negative definiteness of Q’ appears to be a little difhcult, and, there is also the question of whether or not this condition is met in the real world. Usually in the economic literature, the comprehensive requirement that every product follows the law of downward-sloping demand (which means that the direct effect of a price change is negative, or, that all diagonal elements of Q’ are negative) has played an important role on sufficiency conditions for negative definiteness of Q’. Yet another exigence has been that matrix Q’ be strongly quasi diagonalized, what roughly speaking means that there must exist some relation between direct and cross price effects such that the direct effects are stronger than some linear combination of the cross effects. Moreover, there is no economic reason on which the assumption of a symmetric matrix Q’ is based. The only justification for it consists on working with integrable inverse demand functions, in order to obtain a mathematical program with the interesting property of having dual variables easily interpreted as the prices under study. It has not always been remarked in related papers that in fact one needs the symmetry condition to operate with such a simplified model; otherwise, prices must also be included as primal variables, so resulting much enlarged models, with the structure of self-dual quadratic programs, as originally proposed by Plessner and Heady]181 (see also [5] and [9]). And perhaps the extra computational effort will not be compensated in view of possibly unreliable estimates of the demand functions. In fact, the problem of demand estimation is probably the major drawback to apply this methodology on empirical studies, in many cases overshadowing questions of symmetry (or negative definiteness) of the matrix Q’. For a practical insight in demand dependence approximation and model formulation in cases of limited information, the reader is referred to Duloy and Norton[7,19], whose experimental work has been carried on a developing country, where lack of data is a particularly grave problem. -At last, the model is inserted within the general framework of the neoclassical equilibrium analysis, and as such it captures some common hypothesis in this kind of academic thought. In this sense, it is assumed that producers’ decisions are made under complete information about future market prices; uncertainty has not been introduced in any way, and for instance, agricultural studies would make no attempt for possible climate conditions. Also it is supposed that there is no increasing returns of scale, so in the agricultural context a bigger farm would not be more efficient than a little one. 2.3 k nonlinear extension with convexity assumptions With regard to data availability and estimation problems, perhaps the mathematical program (3)‘-(S)’ consti-

256

H. PACCALOUREIRO LUNA

tutes the most adequate structure for empirical studies on spatical models of sectorial economic equilibrium. But in order to discuss the broader context into which the same framework may be used, either at the conceptual level of mathematical economics, or from the viewpoint of large-scale mathematical programming, we present here-appealing to convex requirements-a nonlinear extension of the economic model represented above. Maintaining the same notation with regard to the variables, a first modification introduced here concerns the regional aggregate input-output functions. At each region r, if production activities are Operated at level xr, then the input of restricted resources are given by the convex function h’(x’) (instead of B’x’) and the output results are expressed by the concave function g’(x’) (instead of A’x’); also it is assumed a convex cost c’(x’) associated with unrestricted inputs from other sectors of the economy. The linear structure of the distribution activities is maintained, so enabling as well the exploitation of classical transportation problems embedded in the model. See Fig. 3 for an illustration of the problem as that of a large system composed by the regional economic subsystems (each of them with a production and a consumption sector), and where the coupling variables are the transportation flows. The regional aggregate demand functions may be nonlinear, i.e. (1)

r= 1,. . . ,R,

q’= y’(p’),

but it is required for (1) to have an inverse demand function: (2)

p’=0’(q’),

r= l,...,

R,

Region r

I b’2h’(x’)

(R~quirwnmts on restrict.d

-I--

l

Production xr

,. .

t

rncrwry

IWII,

subject to (4)

r= 1>..., R

i--

h’(x’) 5 b’

(5)

Analogously, one has the following compact formulation:

I

9

0(q) . dq + c(x) + Sj subject o (3) Minimize S.X’=OJ~O ” q-g(x)+

(4)

TfsO

h(x) I b

(5)

where Jo40(q). dq and c(x) are the results of adding respectively $o”‘0’(q’)dq’ and c’(x’) for all regions, g(x) and h(x) are correspondently partitioned on g’(x’) and h’(x’) for r=l,..., R; and 8, b are the same vectors as that defined before. Together with the above stated primal constraints, if one assumes differentiability for c(x),g(x) and h(x), then the following optimality (Kuhn-Tucker) conditions are necessary and sufficient: (2)

p’ = 0’(4’)

(6)

Vc’(x’)

(7)

p’-p”s

+ u”Vh’(x’)

t p”Vg’(x’)

d’: for all set (r, fl of interregional arcs

(8) p” 2 p” - d”, for the set (r, o) of exportation arcs

I I

Regional supply f’ P’(X’1 fro n

.

Multi-commodity ‘low*

from

I

(’ 1”

_._ __)

C’(g)

(From, -107

-h’(x’) --

/

foiFrom abroad

TOr

I

(9) p” 5 p” + d”‘, for the set (0, r) of importation arcs

I fC\

inputs)

q’-g’(x’)-~E~~r)VY“‘)~o

1

**ctors -. --* C’W

Resource avoilobillty

gr(q’) dq’+ c’(x’) + dj 1

r=! 1..., R

and yet another exigence is that one can integrate (2) resulting a concave integral function. Or, we must work with a demand function having its gradient matrix-at every point where it is defined-symmetric and negative definite.

Inputs from Other

Thus it is defined the following convex programming problem:

x’(A vector where each cCW”po”e”tx7]gives the operotionol level Of Mtlvlty 1 in regkvl r )

‘4’( Vector wllh each corn wnmt qT raprrrents r* Of

> glonalconsumption COmmodity I) From r

Fig. 3. Sketch of the national system and the regional subsystems related with a sector of the Economy.

Planning and mathematical

programming

the above conditions expressing the relationship among the gradient vectors of the objective function and of the constraints; and the slack complementarity conditions being: (10) [Vc’(x’) + u’Ph’(x’) - p’Vg’(x’)]X = 0, r=l,....R

(11) (d”+p’‘-p’)f”=O,

all

(12) (d”-p”+p”)j”=o,

all (r,o)

(13)

(do’ + p” - p”)fol=

0,

(r-r3

all (0, r)

(14) P’~[q’-8’(x’)-~~~‘~(rv’Y]

=O. 1

(15)

u”[b’-h’(x’)]

=o

r=l ,...,

R

I

where Vc’(x’) is the gradient of c’ at x’ expressed as a row vector; Vg’(x’) is a matrix whose nth row is the gradient of g.’ at x’; and Vh’(x’) is also a gradient matrix having a similar definition. The primal restrictions (4)-(5) together with nonnegativity constraints and the other Kuhn-Tucher conditions (2) and (6)-(15) reflect a slightly more general situation of spatial competitive equilibrium in an economic sector. Some of these relations are identical to that established on the anterior subsections, as it is the case of (7)-(9) and (ll)-(13) which express a locational price equilibrium (or stability on the competitive behavior of the distribution agents). But now (2) fulfils a nonlinear econometric dependence between the regional prices and consumers demands. As before, (4) assures no excess demand for every region, and its slack complementarity (14) permits a positive regional price only if there is no excess supply of a product; the only difference from (4)’ and (14)’ is that now both (4) and (14) have inside it a concave differentiable output function, instead of a linear one. Here the constraints (5) maintains the regional activities within a convex production possibility set, and from (15) one can impute a positive price only for a scarce resource. Finally, (6) and (10) stipulate the marginal conditions on equilibrium for producers: at every region, the marginal cost of any activity (resulting from summing up the marginal cost of inputs from other sectors plus the marginal value of limited resources) would not be smaller than the marginal value of the output, and yet an activity would be made only if its marginal cost is equal to its marginal revenue. That is to say, (6) asserts that no production activity permits a marginal profit in equilibrium, and (10) states that only activities which breakeven are operated at a positive level in a competitive equilibrium. In the same way, all the other economic properties discussed in the previous subsection for the simpler model are analogous for the actual nonlinear extension, so we make no further comments on these points.

257

to spatial price equilibrium models

given commodity prices. But at the superior level, the task of coordination is quite different in each of these two procedures: instead of making a simple updating of prices as guided by the last observation on excess demands, the Dantzig-Wolfe’s method accumulates all the information previously generated and so combines efficient production responses in order to find a possibly better structure of prices. Thus the Dantzig-Wolfe’s process requires much more work at the coordination level, but it is much more robust in the sense of assuring convergence to optimality. Moreover, if for a given optimal price there are many alternative good solutions at the subsystem (or subproblem) level-[note that uniqueness at this level can be practically assured perhaps only with requirements of strict convexity as studied by Lasdon [ 13]]-then a Dantzig-Wolfe’s procedure is able to impose for the subsystems optimal solutions which also correspond to optimality in the overall system. In this section we shall discuss a variant of the Dantzig-Wolfe’s decomposition algorithm applied to the mathematical program (3)-(S) or to its simplified version (3)‘-(5)‘.We are mainly interested in a variant that could eventually be applied within the context of a decentralized planning, such that at the national level only partial information would be necessary concerning the behavior of production and consumption at each region (namely the regional excess supply or demand associated with alternative prices). The master program would be computed at a national planning agency and the subproblems might be solved offline at regional bureaus of study; alternatively, for some region where there is no mathematical model, the solution of subprograms might be subtituted by conjectural studies on possible regional responses for given prices. Such a possibility arises if, for each cycle, it is conveyed from the master level a batch of alternative vectors-instead of only one coordination vector-to each subproblem which is solved many times so as the number of proposed alternatives, a set of possible responses finally being transmitted from each region to the national coordination. In as much as quasi-optimal solutions might be found in a practically reasonable number of cycles, a procedure of this type could overcome one of the major drawbacks on guiding information for planning as suggested by a decomposition method. For its own sake of enabling a decentralized planning with a practicable repetition of the center-region’s dialogue, it is expected that the advantage of a few good cycles calculation in some cases can even compensate an eventual increasing in the total computation time-with respect to a global strategy or to using standard cycles as the original decomposition method that inspired such a procedure. 3.1 Problem manipulation For the sake of obtaining a further simplified notation, the following auxiliary functions may be defined:

3. TWO-LEVELFUNNING WITE PRICE INDICATORSAND DANTZK-WOLFE’S DECO-ON

As introduced in Section 1, the tatonement process is the classic price-coordination scheme one encounters in the economic literature. Also the Dantzig-Wolfe’s decomposition algorithm has suggested price-directive hierarchical procedures for planning. Usually both these classes of multilevel schemes lead to the same type of subproblems at the inferior level, mainly in what concerns maximization of profit in production units for

(16) w’(q’, x’) = -

‘Z’ I0

V(4)

(17) e’(q’, x’) = q’ - g’(x’),

dq’ + c’(x’),

r=l,...,R 1

where (16) accounts for a sort of “regional-sectorial discounted cost” given by the production cost of inputs from other economic sectors less the regional welfare accruing from the consumption of the sectorial products:

258

H. PACCA LOUREIRO LUNA

(17) is the vector of “regional excess demands”, each component being given by the difference between consumption and production of each commodity (if it is negative there is a regional excess supply of the commodity). Therefore, the problem (3)-(5) may be synthesized in the following formulation: w(q, x) + Sf subject to

e(q,x)t TflO where S = S’xS’x . . . XSR = ((4, x) q = (ql’. . .q”. . .qR’), q’ contained within sufficiently large bounds, x = is a (x"...x". ..xR')',x'z 0, h'(x')sb', all r}; w(q,x) real value given by Ew’(q,,x’) which follows from , summing up (16) for all r; e(q,x) is a vector whose partitions are given by (17) for every r. An interesting specialization of the Dantzig-Wolfe’s approach for (18) corresponds to approximate w and e by inner linearization over an arbitrarily fine base of points {q’, x’}E S, so resulting the following linear master uroblem:

over slack or weight variables A,’ in determining which variable is to enter a basis. It follows that selection of a column associated with a new basic variable At’ will be made only if prices satisfy conditions (7)-(9), since otherwise a distribution flow variable would have entered the basis. Another consequence is that no slack variable will be on basis with such a priority convention if (8~(9) limit prices between positive bound values. Remark also that under these conditions (20) corresponds to the usual optimality test of stopping computation when a minimum relative cost factor of non basic variables is nonnegative: indeed, the column associated with A,’ is

cost W’(#, X”) _----_-__-

simplex multipliers _---------

coefficient

0

-P'

e’W, x”)

-

0 0

r

C,2 h,‘w,’ + Sf

lbji$-~oe (19) . z:L’ &‘=

r

I

,I r=l ,...’

en - ~&Ffra)~O I

1

III

i’

i

R

J

where w,’ = w’(qn, x”) and en = er(qn, x”). The following theorem may be stated with base on Dantzig[4] (see also [12,13]). i%orem: Let (-p”;~“)=(-p”‘...-pR”;~l’...~R’) be simplex multipliers corresponding to an optimal solution (A';f") = (A I"...AR"; f") of the restricted master program (19) at cycle 1. If

0

YIR ___------_

_-------

and pricing out this column yields the relative cost factor 8,‘= w,‘++p”e”-7, 4’1

= c’(x”) -

I0

@‘(q’) . dq’ + p”q”

-g’(x”)

The problem of finding a point (q”, x”) E S’ minimizing the relative cost factor C+’ for given (p” y,), becomes the rth subproblem: c’(x,) - p”g’(x,) + p”q’-

(20)

Minimum (q’.x’ES, for r = 1,. . . R

{w’(q’,x’)+p”‘e’(q’,x’)}-~~,O,

I

r= 1,...,

xA,'xn t

h’(x’)

s

.” 8,(4’) dq,

br

xRf’;f”)

q* = TA,‘q” x’=

I

(21)

subject to

then the point (4”; x”; f”) = (4”‘. . . qR”; xl”. where

R

is an optimal solution in problem (18) [or in original versions (3)‘-(5)’ or (3)-(s)]. The proof of such a theorem is a straightforward extension of the original proof developed with inner linearization on the whole space of primal variables[remark that here the sub-space of the distribution flow variables is not included in the inner linearization process]-so we do not prolong the text with a similar reasoning for this case. The solution of the master problem (19) can be made by the simplex method with the following priority convention: distribution variables fnnr must be given priority

- lln

Regional Subproblem The regional subproblem can be separated into two other simpler problems: -The first is a restricted convex program (which is linear if the production relations are linear) and which reflects a maximum profit behavior when producers are faced with prices p’: Mi:jpize (22)

c’(x’) - p”g’(x’)

subject to h’(x,) 5 b’.

-The second is an unrestricted whose minima1 solution computation evaluation of the demand function:

convex program requires a simple

Min$Gze ( p ” q ’ - joq’ V(q’)dq’).

Planning

and mathematical

programming

The minimal solution must satisfy the inverse demand function p’ = Q’(q’), so the optimal q, is the consumers’ response for the price p,, given by the direct function (1)

4’ = Y’(P’).

3.2 Solving the master and generating prices with transportation problems The dual of the restricted master problem (19) is: (23) (24)

Maximize cl/, ll.P”O r

Minimize u(y) + ue(ye) I Y.P (25) Subject to y + ye 5 0,

all (r,P)

(7)

p”-p”
(8)

p”rp”-d”

for (r,o)

(9)

p"sp"+d"'

for (o,r).

This dual problem may have an interesting interpretation. The constraint

economic

y E Y,

where Y = Y’xY*x.. . xYR ={Y/Y’ = (Y”, . . . , YR’), for each r = 1,. . , R there exists (q’,x’)E s’ such that e’(q’, x,) 5 y’}; ue(ye) = min {SflTf 5 y@, f ~0 and limited above by a sufficiently large vector} accounts for the minimal distribution cost when node injections are bounded above by ye; u(y) = min{w(q, x)/e(q, x) 5 y, (q, x) E S} is the minimal sectorial discounted cost when excess demands are bounded above by y.

- 1’” g’(q’) dq’+ c’(x”) + pr’en 2 q,+ O-consumers’ welfare

259

models

demands). The solution of these problems provides a sequence of spatial prices (dual variables), and it converges to a vector of prices corresponding to optimality in the restricted master problem (19). More preciselly, the original problem (18) is equivalent (see [ 161) to the product-allocation problem

subject to

w,’ + p,’ e” 2 (, for every r and all t

(24)

to spatial price equilibrium

a national subsidy (if negative it is a taxe) applied in region r

production cost

I cost of supplying the excess demand (if negative, it is the benefit of the regional excess

SUPPlY)

restrict the value of a subsidy to be never greater than the production and excess demand costs minus the consumers’ welfare. An optimal solution accounts for a subsidy yi (or a differential value to compensate location) which exactly maintains a sort of equilibrium costbenefit on the regional activities related with the economic sector under study. The conditions (7)-(9) doesn’t permit a positive profit for any distribution agent, so the solution set of this dual problem contains structures of values for which benefit is not superior to costs either for any distribution activity or for the regional production-consumption activities indexed by t. Under mild assumptions it can be shown that the above problem corresponds to a relaxation of a tangential approximation addressed to the dual of the original problem (18) (see [13,16,20]). In order to refine this approximation with new cutting planes (24), or, in order to introduce a new set of columns in the restricted master problem (19), a series of transportation problems may be used to generate spatial prices, so enabling a multiple pricing out operation solving many times a minimum reduced cost subproblem (21). Such a procedure is consistent with the convention of priority on distribution variables to enter a basis, giving spatial prices compatible with equilibrium on the distribution agents behavior [conditions (7)-(9) are satisfyed together with its slack complementarity]. The choice of the price generator transportation problems could be guided by a primal decomposition method applied to problem (19), since such a technique provides a series of transportation subproblems with different right hand side (supplies and

Also a similar primal decomposition approach may be addressed to the restricted master problem (19)-which is an arbitrarily fine inner approximation of (18)considering two principal divisions: -One partition corresponds to the transportation flows and can be separated into a series of simple distribution problems (one for each class of goods) given by Mir$i

(26)

,z

I

n

d.“f.* + p.O(T fnD’- T fn-> subject to

-a~c’jUnar-fnrQ)~y,,‘?

r= 1,2 ,....

R

where (_f)” is the vector of flows of product n, each flow being nonnegative and limited above by a very large number; -Another partition corresponds to the consumption and production activities and can be separated into R linear programs (one for each region) given by Mi$mke (27)

A EA

~AI’w~’ subject to ’ zA,‘e”ly’

I

where A’=

I

A’lA’~O,

XI\:=1

t

I

.

260

H. PACCALOUREIRO LUNA

Since problem (19) is an inner linearization of the original problem (18), the same primal decomposition approach applied to (19) corresponds to a product-allocation problem contained within (25). Also since our latter purpose is to solve (18) [or equivalently (25)], if one is interested to do a tangential approximation in the product-allocation problem equivalent to (19) then it seems adequate the use of outer-linearization to approximate Y’. Here Y’ = Y”x.. . xY”x.. . xYR’, where Y” is the set of maximal regional excess demands y’ for which problem (27) is feasible with the columns e” existing at cycle 1. Remark that Y’ C Y, so a containing polyhedral approximation of Y’ may lead to an allocation vector y such that yg Y’ but y E Y, and so the use of spatial prices related with such an allocation (and generated by correspondent transportation problems) may eventually enable subproblem (21) to generate “good” columns to the restricted master (19) of next cycle. In order to update the allocation vectors the following coordination problem may be solved (see [ 16,131):

this problem by the strategy of relaxation [12,161. Violated constraints could be generated as needed by performing a phase one operation on the constraints of (27) for a fixed y’: a suitable extreme ray (r!, P”) may easily be found from the simplex tableau if the system of equations is infeasible [13]. The constraint formed from this is added to (28) along with similar constraints for other regions r and previously generated constraints (29) refining the tangential approximation of Y’, SO being obtained a new coordination relaxed problem (28) which updates the product allocations vectors y, ye. A final simplification concerns the fact that if (y, y”) is optimal in (28) then (y, - y) is also optimal in (28) since it is also feasible and it implies a c@ value not greater than that related with the original pair (y, y”) [remark that the pk are positive]. We always use the (y, - y) type of solution in (28) so we simply tell about the allocation vector Y.

3.3 Aigotithm and planning process The techniques discussed on the two previous subsections are used in this subsection to specify an algorithm to solve problem (18) [or its original versions (3)‘-(5)’ or Minimize a, + ue (3)-(5)]. The algorithm is presented together with an Y.Y00J%4 economic interpretation suggesting a decentralized proSubject to y + ye I 0 cedure for planning. Remark that from the computational (28) ( al2 v’(Yk)-pk[y -y”l point of view it might be devised three calculation levels: the first level solves problem (21) the second one k=l,...,v =~{~“(Y”C)-p*[Y’-Y~l} computes problems (26) and (27) and finally a third level solves a relaxed (28) with the constraints (29). Instead, ae 2 ue(yek) - pk[y8 - yOk]yE y’ I from the viewpoint of economic planning it is conceived only two principal levels each of them being amenable to where the index 1 denotes that (28) is associated with the solution by a different computer at different places. The restricted master problem (19) existing at the central same inferior level of solving many subproblems (21) for planning at the Ith cycle of information flow between the every r is supposed to be associated with regional national and regional levels; pk = (p”, . .pti,. , pRk) bureaus of study, each of them having its regional model is the vector of dual prices related with the consumpwhich enables prevision on consumers and producers tion-production partition obtained by the R linear pro- behavior for given regional prices. A central planning similarly pk = agency at the superior level receives information on the grams for (27) Y=Yk; (p’k,...pnc,. . ,pRk) is the vector of dual variables regional excess demands and solves the restricted master associated with the transportation partition and obtained problem (19) providing a new sequence of regional by solving (26) with node injections ynBti for every n. prices. This work includes the second and third above Note that the distribution subproblems are feasible for specified levels of the hierarchical computation strategy; any set of node injections y”, but the excess demand in fact, problems (26)-(28) result from a primal decomvector y must be contained on position method applied tdo (19). Within this context, the following procedure is applicable: Y’ y/there exist h’r 1

0,

CA,’ = 1, I

zhlle*>y’, t

r=l,2

,..., R,atcycleI

Initialization: At each regional bureau Step 0: Having information on the bounds and a first estimation of commodity prices at region r, compute a

I

or by the elementary theory of linear programming [12], y is in Y’ (or y’ is in Y” for every r) if and only if ui-p”‘y’sO,

(29)

all j

where (u:‘,p’) is the jth generator of the convex polyhedral cone: C: = (/.~r,p’)](1 1 . . . l)‘/~, - E,“p’ I 0}, I

El’ being the excess-demand submatrix (with columns e”) existing for region r at the Ith cycle of solving the master problem (19). Because only a small proportion of the linear constraints (29) will probably ever be binding at an optimal solution of (28). it would be natural to solve

series {g”} of supply responses to commodity prices (p”}. It is done by calculating g’(x’), where x’ is an optimal solution in (22) for pr fixed in the following sequence of values: t = (1) the regional estimated price for every component (class of goods), t = (2) the maximal regional price for every class of goods, t = (3) the minimal regional price for every class of goods, t = (4) the lower bound for the principal regional product and the maximal price for any other class of goods, t = (5) the upper bound price for the principal regional product and the minimal price for any other class of goods, any (t) any convenient vector of regional prices obtained by fixing each component equal to one among three values: the maximum, the estimated or the minimal commodity price. Also a series {qti} of demand responses to the same alternative prices (p”} is known with base on the

261

Planning and mathematicalprogrammingto spatialprice equilibriummodels demand function q’ = y’(p’) for each region r. Doing e” = $‘- s* and wtl = -104” 0’(q’) dq’+ c’(x”) calculate the regional series (w”, e”), and send this information to the central planning agency. [The procedure continues at step 3 with v = 0, y’ = q’ - s’ where q’ and s’ where generated for t = 1; use t = 5 and t = 4 to find respectively a lower and an upper bound for the allocation yn’ of the principal product n in region r, and stipulate convenient initial bounds for other components of the vector y, defining Y”.]

If the primal is infeasible for any region, the procedure returns to step 2 with new cuts generated by the extremum rays (pi, p”) associated with every region r for which such an infeasibility occurs. Otherwise, recover an optimal solution and an optimal multiplier vector p’ for every r: compose the vector p”+’ and also name p”’ the vector of spatial prices currently obtained by the transhipment problems. Step 4: Test the following termination criterion for cycle I at the higher level: if

A standard cycle. At the central planning agency Step 1: If there is no negative relative cost factor

(31) ~~~~~{u~(yl’)+ue(-y’)}-(u,tu~)“+’~E

associated with the optimal multipliers of the lth restricted master (or, if w,’+ p”’ e’ - r,’ z 0 for every region r), terminate with the optimal solution (q’, x’, f’; p’). Otherwise, enlarge the restricted master with the regional coefficients (w,‘, e”) generated in the previous cycle, increaseIbyl,puty’=y”“,v=Oandgotostep3. Step 2: Find a new allocation vector y which minimizes the presently best underestimator of the objective function in (19). It is done by solving Minimize U, + a0

then y”’ is sufficiently near optimal in the productallocation problem associated with the current restricted master of cycle /. In this case, compute W*’= T h”w’ r=l ,..., e’* = ?A” e” I

subject to

YEYo.qog

2 O’(Yk)- pkIy - Y’l u%¶BuvB(-yk)-pk[-ytyk]

w

R,

by using the optimal A at step 2 for each region r, overestimate a subsidy which assures a benefit not inferior to cost on the sectorial activities

k = 1,. . . ,Y

n: = w.’ t pd’ e’* CL:’ - pO”y’5 0. for each r and all j generated at cycle 1. Step 3: With the allocation vector y = y’+’ calculated above solve a series of transhipment problems (one for each class of goods) and also a series of linear programs (one for each region): -each transhipment problem is given by the pair

where p’ is the regional subvector contained in p”“. Send for each region r the subsidy ll:, the vector of values p’ and every other vectors of regional-prices p” contained in the series Ip’} generated at step 3 in the present cycle.

Primal

Dual

Minitize (d)“Cf)“, subject to (30)

P&(f,“-f~m)=~~,

r=l,...,R

pf”” - fn”‘) = yno where cf)“, (d)“, (p)” are sub-vectors with components referring to product n and y.“ = -By,’ is the level of international flow. The dual solution for this problem must be selected with pnOfixed at the international price level, so obtaining a new set of spatial prices which is appended to the series b’} generated in the present cycle at the central agency. -The linear program associated with region r is given by the pair Primal

Minimize x&‘w,’ *,‘ZSJ ,

I

Maximize pnOynot cp.‘y,’ subject to I (I)” p$ - p,,’ 5 d,*+, for all (r, 7)” pnr z pno - d.“,

for (r, o),

pnr c: p.O t d,“‘,

for (0, r).

On the other hand, if the above termination criterion is not yet satisfied, then increase Y by 1 and return to step 2. A standard cycle. At each regional bureau of study Step 5: For each vector of regional prices p” received from the central planning in the present cycle verify the Dual

subject to

M;‘x$n~ - y”p’ + pn

subject to

CL,- p” e” 5 w”, all t

262

H. PACCALOLIREIRO LUNA

consumers’ demand q” = y’(p”) and solve the convex program which reflects producers’ behavior

calculations are to be made in a central computer then a most efficient alternative might be selected.

(22)

4. TWO-LEVEL

Minimize c’(x’)X’>O

p”g’(x’)

subject to h’(x’).s

b’,

determining an optimal solution x” and a supply response s ” = g’(x”). Calculating the excess demand en = qn - s” and doing w,’ = -Jo”” @‘(q’) dq’+ c’(x”) for all t of cycle 1, determine a new regional series {w,‘, e”} whose last element is (w”, e”). Send this series to the central planning together with information if wr’+ p”‘e’ -Y:
2’) (Aq’, Ax’) + SAf

Ve,‘(?j’, f’)(Aq’, Ax’) - s,’ 5 0, TAf - s“ 5 0 (33);

Ax; 50,

PLANNING WITA DISTRIRUTION TARGETS AND

GROFIXION’S PRO.JECTION/FEUIRLRDIRFCTIONS TR4XNIQUR

For the sake of suggesting a decentralized procedure for planning with targets on the spatial allocation of final commodities, a primal technique of large-scale mathematical programming may be devised [12,13, 161. Usually this kind of technique has the practical interest of generating a sequence of improved feasible solutions. Also if it is applied to problem (3~(5) it can take full advantage of the specific separability of the model, providing many small convex sub-programs and simple network flow problems by partitioning only two principal divisions: the consumption-production part and the distribution part. A large-step subgradient approach to (25)-specializing Geoffrion’s algorithm as proposed in Section 4 of [16]-seems to be a primal decomposition method particularly suitable for a two-level planning guided by the computation of (3)-(j). In addition to the now discussed properties of interest and to an expected numerical effectiveness as being a method of feasible directions, such a technique tends to diminish the necessary number of cycles of information flow between the national and the regional levels in as much as subproblems are solved many times at each cycle. 4.1 The direction-finding master problem Using Geoffrion’s remark [ 161 our master problem can be interpreted as a direction-finding problem addressed to the equivalent program

(32) i

Minimize w(q, x) + of 4.X=0.I~o:Y

subject to

h(x)lb,

Tf~y”,

e(q,x)Sy,

y+y@~O,

which is now a problem with ‘coupling variables’ instead of one with ‘coupling constraints’ [here w(q, x), e(q, x), y, ye are defined as on the previous section]. If a feasible solution (4, f, fi 7) is known, the algorithm searches for an unn_ormalized displacement (Aq, Ax, Af) away from (4, 2, f) and a normalized displacement s away from j such that the initial rate of improvement of the objective value is maximum [as we see below here again the vector y is enough to characterize the spatial allocation, since until optimality it is possible the use of solutions with ye = - y]. Indeed, if f is interior to Y, (4, a) is any optimal solu_tion of the consumption-production part with y = y, f is any optimal solution of the distribution part with ye = jr” = - y, assuming a constraint qualification on the set {x/x 2 0, h(x) I b} and w(q, x), e(q, x), h(x) continuously differentiable, then the direction-finding master problem is (see [13,16]): subject to

r = 1, . . , R, n

such that e,‘(@‘, ~7) = h’,

[since T{= y”]

i such that xi’ = 0,

Vh,‘(f’)Ax 5 0, r = 1, . . , R, m such that h,‘(Y) = b,’ Afn” 2 0, A,” 2 0, Af,,O’2 0 for every arc (L, 7)., (r, o), or (0, r), such that respectively f.” = 0, f-nm = 0 or Lo’ = 0,

k S+S’~o

[sincebyconstructionjrtjjB=O],-l~s.’~l,

-lss.“
allrandn.

Planning

and mathematical

programming

This problem admits a further simplification in that if Af; s, se) is an optimal solution to indicate changes away from (4, 2, f, 7, ye) and if y” = -jr, then se 5 --s together with the second feasibility constraints imply

to spatial price equilibrium

(Aq,Ax,

Af
(22)’

i

y’ = 4’- A’Y that is send to the central planning together with information on:-all the matrix A’;-every row tn of the matrix B’ corresponding to a scarce resource with respect to 8’ [i.e. such that equality holds on the mth constraint in (22)‘];-the index of all non-basic production activities. Step 01: At the central planning agency, solve a transhipment problem (30) for each class of goods, doing y,’ = y,,’ calculated above at each region, and, fixing the international flow at the level y,‘” = -Xy” (it refers to r exportation if it is positive and to importation if negative). Fixing the known value of the international price for each dual variable p.” it is obtained a convenient structure of spatial prices p and distribution flows f to start the procedure.

SO maintaining the equality yB = -y during every cycle. Yet recall that for a fixed vector of node injections ye the (multicommoditylseparable) distribution subproblem

Sf

subject to B’x’ I b’.

It results a spatial allocation subvector

-ss,

(Y,YB)=(~+PS,~e+pSB)=(~+ps,-Y-ps)=(Y,-Y),

Mi;i$ze 2

263

Mi;j&ize (c’ - $A’)x’

and so s’ = -s also corresponds to an optimal solution (Aq, Ax, Af; s, - s), since it is also feasible and it has the same objective value. Starting with node injections equal to the negative of upper bounds on excess demands y” = -j one can always change these values in such a way that

(34)

models

Tf I y’

subject to

must have an optimal solution with f such that 7”= ye, since any feasible solution for which a strict inequality

A standard cycle Step 1: At the central planning, with base on a spatial_ allocation j-related with known distribution flows f and with demands 4 and production activities j computed at the regional level and from which is known the index of non-basic activities as well as the index and row coefficients associated with tight demand targets and scarce resources-solve the direction-finding master problem:

a& cf.“’ - fn”) < yne’ holds for some r, n,

can be improved either by diminishing some interregional flow f,,” or by increasing some exportation flow fn” until equality [remember we assume positive interregional costs

- Pn’Aqn’ t ccr’Ax; j

Aq,’ - C a LjAxi I

i

s,‘,

r= l,...,

R,

1

+ 8Af subject to

n such that 4,,-~a~,&‘=~~‘.

1

TAf I s, (33)’

Ax: 2 0,

j such that %’ = 0,

XbLjAxj’sO,

r= I,.. . , R,

~TIsuch that C b :>j4’ = b,‘,

A>‘,” z 0, Af.” 2 0, Af”y z 0 for every arc (r,‘&, (r, 01, or (0, r), such that respectively f.” = I), f_n’O = 0 or f;l”’ = 0,

- 1 5 S”’ 5 1, and sufficiently high international prices such that the vector 8 has negative components for exportation arcs and positive components otherwise]. 4.2 Algorithm and planning process Using the above discussed properties and applying the algorithm to the linear-quadratic problem (3~(+-in order to do a more complete and practical economic interpretation with fixed input-output coefficients-the following hierarchical procedure for planning may be devised: Initialization Step 00: At each regional bureau of studies, knowing a first estimation of the commodity prices (a vector p’> 0), determine the consumers’ response by the direct demand function 4’ = qs’+ Q’e’, and, generate a possible supply response Al’ with ff’ being any optimal solution in SEPS Vol. 12. No. 5-D

whose economic interpretation is clear:-the targets on the spatial allocation of final commodities (the vector y of maximum excess demands) must be changed towards a normalized direction s whose marginal cost (production plus distribution) discounted by the marginal consumers’ welfare is minimal; in order to maintain local feasibility the associated displacement (Aq, Ax, Af) must correspond to a marginal excess demand limited above by s.’ if for region r and product n the target 6’ is tight: yet, the marginal change on the distribution flows Af must be compatible with a new spatial allocation, as well as changes on the production activity levels x can not increase the use of any scarce resource m in any region r, all the while maintaining nonnegative activity levels. If s = 0 belongs to an optimal solution for the problem (33)‘, then no improving feasible directions exists, so j is optimal in (25) and the procedure terminates with the economic equilibrium solution (4, 2, f; r?, ti) provided by the lately computed values for demands, production

264

H. PACCALOURIEROLuna

activities, distribution flows, commodity prices and resource imputed incomes; [(4, ff; P) are disponible at the regional level and fl; p) are known at the central planning level]. Otherwise, any basic optimal solution for (33)’ can provide a ‘good’ improving feasible direction (s, - s) for (25) at (jr, - jr); inform the sub-vector s’ for each region r: [if multiple optimal solutions exists in (33)’ then alternative search directions may be informed to the regional bureaus]. Step 2: At each regional bureau, for given direction s’ use parametric programming techniques to solve the following quadratic program with changes on its righthand-side: Minimize - 4” psr + 5 P’q’ + c’x’ subject to q’.l-‘Xl > (35)’

q’-A’x’sy’+ps’ B’x’ 5 b’.

Perform the computation increasing /3 by small increments above 0, all the while maintaining optimal solutions for the problem, until for p = Prmax is reached a bound on some ynr. Communicate to the central planning the objective values expressed by the function u’(y’) defined for every y’ = 9 + @s’ for fi within the interval (0, /3,,,,,,J. Sfep 3: At the central planning, determine a step size p for each direction s by solving the following one dimensional restricted version of (25): (36) where each function n,(y’) has been informed from each region r, and, oe(ye) for ye = -j - /3s is given by a parametric programming addressed to (34). [Remark that the evaluation of u”(y”) can be made by additioning for every commodity the optimal value in (26), what uitimately may be obtained by a parametric solution of one transhipment problem (30) for each n and with each regional excess demand changing within the range y,’ = h’ + ps,’ for 0 < fi < min,@,,,}.] Choose the search directions s with step size /? f_or which the optimal value of (36) is minimum; use jr+/% to update f and recover an associated minimum cost distribution flow f and spatial prices p. Send each subvector ji’ to the correspondent region. Step 4: At each regional bureau, recover an optimal solution (Q’, a,) for (35)’ with the right-hand-side jr’ limiting excess demands. Inform to the central planning: -the index of every product n whose excess demand target is saturated; -the index of every scarce resource for the current production level ff’, also informing the correspondent input coefficients if the resource was not scarce in previously studied solutions; -the index of all zero-level production activities. [The procedure returns to step 1.1 A first point of explanation concerns the use of Everett’s theorem (e.g. see [13]) to initialize the algorithm with a feasible convenient spatial allocation y. In fact, in the general case, determining an optimal f’ for (22) and taking 4’ = $I@-‘) corresponds to solving the (separable in q’ and x’) Lagrangean problem

0’(q’) dq’

I

+ [c’(x’) -

/7”g’(x’)l

subject to h’(x’) I b’, which maximize consumers’ surplus+producers’ profit for the estimated regional prices p’>O. Thus the theorem assures that (4: jr) solves the modified primal problem MJGE~

-

I

0*’ 0’(q’) dq’ + c’(x’)

subject q’ - g’(x’) I jr h’(x’) I b’, where f’ = cf’-g’(?‘), which is just the allocation vector used to start the procedure. The theoretical background evolving the master problem (33) requires ji~int Y. Remark that at step 2 one stops increments on the step /3 when is firstly reached a bound on some y.‘. So at each region r it is supposed known an upper bound on the excess demand ynr of each product n, which may be given as a result of the most pessimistic consumers-producers’ response when the regional prices are minimal for product n and maximal for every other product; on the contrary, it is known a lower bound resulted from a consumers-producers’ response for a maximal price of commodity n when every other product has a minimal price. Since to define Y the vector of demands y is contained within as large as desired bounds, then the algorithm always maintains jr E int Y. The same interiority property is also valid for any ye, since the distribution part is feasible for any node injections if one relaxes upper bounds on flows as in the original model. We have tried to suggest a specialization with ..a low number of cycles of information flow between the national and the regional levels. In this sense, not only one but some directions of changing the spatial allocation might be used at each cycle, provided problem (33) presents alternative optimal solutions. On the other hand, in order to diminish the mass of information to be conveyed at each time, the above algorithm requires two inter-level communications to perform a complete cycle. Although in some cases the central planning might require stored information on all the regional inputoutput matrix [A”, B”], note that in this planning procedure all the analytical and computational burden of solving the parametric subproblems (35)’ should be made at each regional bureau of studies. In the linear-quadratic models it is quite important the resulting simplificationfixed input-output coefficientsin view of reducing the mass of information to be transmitted each time. In the general convex case the central planning could not work with a fixed data-base [A’, B’] and receive pointers (index) of columns necessary to solve each new masters; instead, the marginal products and the marginal resources change for different activity levels and all its necessary columns must be informed each time. A final note on the assignment of tasks is that here the transportation problems are supposedly solved at the central planning level, together with the solution of any step size problem (36). From the organizational viewpoint, the solution of these problems might be conceived in the lower level at a commercial studies organism, which

Planning and mathematical programming to spatial price equilibrium models receives information on the regional excess demands and on directions of its change, determines the associated minimal distribution cost (discounted by exportation earnings), an informs each time:-either the onedimensional function v”( .) of B included in (36), or, -an efficient solution of distribution flows f and spatial prices p associated with a certain allocation j. At last remark that both the regional subproblem (35)’ and the master problem (33)’ are specially amenable to decomposition techniques in order to be solved. At a first glance we see that subproblem (35)’ becomes much easier after a dualization with respect to the excess demand constraints, since it leads to a Lagrangean problem (21)’ whose separability for fixed prices p’ rules out difficulty on the quadratic partition by determining the optimal 4’ with a simple evaluation of the direct demand function. Under the stringent condition of a strictly convex cost function c’(x’) the dual function would be differentiable and an efficient computational method for solving the dual (e.g. in [ 131) might be applied; in a more general or linear case tangential approximation of the dual function (the Dantzig-Wolfe decomposition) should be recommended, if possible together with parametric programming techniques. On the other hand. the master problem (33) seems quite suitable to be solved by some technique of projection, by fixing temporarily the direction-vector s and solving many transportation subproblems (one for each commodity) and many linear programs (one for each region). For this purpose this purpose the Benders’ decomposition technique seems to have a practical interest in that its coordination problem is a relatively simple linear program with bounded variables s,‘; moreover its requesting use of the lately generated spatial prices enables economic interpretation of possible significance for the planning process. 5. CONCLUSION We have studied specific variants of two algorithms of

mathematical programming decomposition applied to a typical spatial price equilibrium model. The computational strategies-giving attention to the separability of the model and with emphasis on the transportation problems embedded within it-have been studied with economic interpretations suggesting hierarchical procedures for planning. In order to explore such a possibility, the proposed methodologies-the first of the price coordination type based on Dantzig-Wolfe’s principle and the second with a net-output target coordination, both using Geoffrion’s primal decomposition techniques-are designed to have a low number of cycles of information flow between levels. Two methods suppose the initialization is made with a first estimation of the commodity prices at each regional market. Such an estimation could be provided by the central planning, by using the spatial prices generated with distribution problems associated with an initial guess on the regional excess demands. Alternative initialization procedures were not discussed here, and there is no attempt to particular strategies which might be more effective to completely centralized computation. Up to date we have no information on substantial numeric experiments with decomposition methods applied to spatial price equilibrium models, and in this way we are only at the beginning. Related remarkable experience-also valuable for its participation on a planning process-is reported in the book of case studies

265

in Mexico (e.g. [7, 141). referring to the partial equilibrium of the agricultural sector and taking different regional productions but only a unit national market. For their model the Dantzig-Wolfe decomposition does not require less computer time than the standard techniques of a modern linear programming package. But their experiment has already shown that in important reduction of cycles of information flow between levels can be attained simply by an initialization that solves many times each regional subproblem, since it permits to start the master computation with a rather fine grid. Partially based on such results, we have been encouraged by the facts that, for models with spatially separated regional markets, adapted decomposition methods might have numeric advantage in that they could explore the highly efficient computational techniques for standard transportation subproblems, and yet since convenient variants of such methods could have a few good cycles calculations it could be straightforward their solution within the context of decentralized procedures for planning. From the computational point of view we have outlined the algorithmic framework within which a further detailed computer implementatton should be made. For instance, a point of important development should be how to carry out an effective computation and information storage during successive executions of the regional and the network flow subproblems. On the other hand, from the viewpoint of the economic planning, in spite of all the difficulties stressed in Section 2. if both model-builders and decision-makers are aware of the real world simplifications implied by this model, it seems to be useful to improve knowledge on the behavior of a spatially distributed economic sector. Moreover, since its computation strategy can be carried with such convenient decomposition algorithms of mathematical programming, it follows related decentralized schemes: either for the hierarchical control of centrally planned economies, or to stimulate the information flow in a competitive economy (in other words, following the decomposition methods, integrated national and regional bureaus of study could collect, analyse and disseminate information about future market conditions). REFERENCES

1. H. Nikaido, Conaex Structures and Economic Analysis, Academic Press, London (1968) 2. H. Scarf, The Computation of Economic Equilibria. Yale University Press, New Haven (1973). 3. P. A. Samuelson, Spatial price equilibrium and linear programming. Am. Econ. Rev. 42. 283-303 (1952). 4. G. B. Dantzig, Linear Programming and Extensions. Princeton University Press, Princeton, NJ. (1963). 5. T. Takayama and G. G. Judge. Spatial and Temporal Price and Allocation Models. North-Holland, Amsterdam (1971). 6. J. Vercueil and L. Farhi, Recherche pour une Planification Coherente: Le Modtle de Prevision du Ministere de I’Agriculture, Editions du Centre National de la Recherche Scientifique, Paris (1969). 7. J. H. Duloy and R. D. Norton, CHAC. a programming model of Mexican agriculture. In Multi-leoel Planning: Case Studies in Mexico (Edited by L. Goreux and A. Manne), pp. 291-

337. North-Holland Amsterdam (1973). 8. H. H. Hall, E. 0. Heady and Y. Plessner, Quadratic programming solution of competitive equilibrium for U.S. agriculture. Am. 1. Agn’cul. Econ. IS. 536-555 (1%8). 9. H. H. Hall, E. 0. Heady, A. Stoecker and V. S. Sposito, Spatial equilibrium in U.S. argiculture: a quadratic programming analysis. SIAM Rev. 17,323-338 (1975).

266

H. PACCA LOUREIRO LUNA

10. J. F. Shapiro. OR models for energy planning. Camp. Ops. Res. 2, 145-152 (1975). II. W. W. Hogan, Energy policy models for project indepen-

dence. Camp. Ops. Res. 2,251-271 (1975). 12. A. M. Geoffrion, Elements of large-scale mathematical programming. Management Sci. 16.652-691 (1970). 13. L. S. Lasdon. Optimization Theory for Large Systems. Macmillan, New York (1970). 14. J. Kornai, Thoughts on multi-level planning. In Multi-level Planning: Case Studies in Mexico (Edited by L. Goreux and A. Manne) 552-55 1.North .Holland. Amsterdam (1973). (5. E. Mafinvaud, Decentralized procedures for planning. In Activity Analysis in the Theory of Economic Growth (Edited by E. Malinvaud and M. 0. L. Bacharach). Macmillan, New York (1967). 16. A. M. Geoffrion. Primal resource directive approaches for

ontimizine nonlinear decomposable systems. 0~s. Res. 18. 375-403 fi970~ 17. H. P. L. Luna, Economical interpretation of Benders’ decomposition technique applied to location problem. Proc. of the IFACIIFORS/IIASA Conf. on Dynamic Modelling and Control of Nationa/ Economies, Vienna, Jan. 1977. NorthHolland. Amsterdam (1978). 18. Y. Plessner and E. 0. Heady, Competitive equilibrium solutions with quadratic programming. Metroeconomica 17, 117130 (1965). 19. J. H. Dulay and R. D. Norton, Prices and incomes in linear programming models. Am. J. Agticui. &on. 57, 591600 (1975). 20. B. C. Eaves and W. 1. Zangwill, Generalized cutting-plane algorithms. SIAMJ. Control 9, 529-541 (1971).