Leblanc, Daniel

Queyranne, Maurice

PolyPublie

Génie IndustrielCOMPARISON OF ALTERNATIVE METHODS OFUPDATING INPUT-OUTPUT TABLES: A NOTEdaniel LEBLANC and maurice QUEYRANNEDEPARTEMENT DE GENIE INDUSTRIELECOLE POLYTECHNIQUE DE MONTREALJUNE 1979Rapport Technique no, EP-79-R-21Ecole Polytechnique de MontréalCA2PQUP479R21 Campus de l'Université de Montréal Case postale 6079 Succursale "A"Montréal, Québec H3C 3A712 NOV 1979ECOLE POLYTECHNIQUE BIBLIOTHEQUECOMPARISON OF ALTERNATIVE METHODS OF UPDATING INPUT-OUTPUT TABLES: A NOTEDaniel LEBLANC and Maurice QUEYRANNE2DEPARTEMENT DE GENIE INDUSTRIEL ECOLE POLYTECHNIQUE DE MONTREALkCONSULTER SUR PLACEjune 1979RAPPORT TECHNIQUE NO: EP-79-R-21COMPARISON OF ALTERNATIVE METHODS OFUPDATING INPUT-OUTPUT TABLES: A NOTE*Daniel LEBLANC and Maurice QUEYRANNEECOLE POLYTECHNIQUE DE MONTREALDEPARTEMENT DE GENIE INDUSTRIELJune 1979.RAPPORT TECHNIQUE NO. EP79-R-21* This research was motivated by a study on the Input-Output table of Québec carried out for the Office de Planification et de Développement du Québec. The authors also thank B. T. Smith for his comments on a first draft of this note.VaAsLoub m&thodi fioA updating Input-Output tablai have, beenIn thisABSTRACT:ptiopoAed and compaAtioni have been made on statistical bases, note me show on theoretical bases that several Linear programming formu­lations are not suitable to this problem since a large proportion of the resulting coefficients assume arbitrarily preset values.1The problem of updating Input-Output tables from an initial tableand known row an column totals is frequently faced by government agenciesThe same problem also arises in statis­tical studies [ 14 ], [ 6 ] as well as in the estimation of telephone traffic [ 7 ] , [ 11 ] , [4] and urban trip distribution [ 5, chap. 4 ] . methods for solving these problems have been proposed (see the surveys by Lecomber [ 8 ] and Stemmeulen [ 13] for the application to Input-Output tables).in their policy-making activities.VariousSeveral comparisons have been made between these methods on anIn particular Davis et al. [ 2 ] haveexperimental or statistical basis.compared the well-known RAS method and two variants of a Linear Programming (LP) method proposed by Matuszewski et al. [9 ] . They have used the 1963US National Input-Output table to derive a 1967 updated table which was com­pared with the actual 1967 data. They concluded that there is a clear su­periority of the RAS technique over the LP approaches.In this note theoretical considerations are used to establish theinadequacy of these LP approaches to this problem. We show how the pro­posed LP method can be formulated as a variant of the transportation2problem of linear programming, the solution of which involves a largeproportion of updated coefficients assuming arbitrarily ptiUeX VCUtuZA.In mathematical notations, the problem considered in [ 9 ] and [ 2] is the following: given an m x n table of reference values x and column totals w^ (i=l,...m) and u^ (j=l,bounds r^ and sjj respectively for cell (i,j) for year t, find values x^j for all cells (i,j) such that xrowij’n) and lower and upper• è •^ 0 whichijI / x2 I (1)minimize Xij ' "U iji > j(x,Jijt tS (2)subject to Xij Uj ’it t2 (3)Xij = wi *jtt t (4)rij ^ Xij ' “ij *sHere I y I denotes the absolute value of y and it is assumed that2 u^ = 2 w!1 .2 i> 0 are considered.Here and in the sequel, only cells (i,j) such thatjt , t .. and s.̂  are given for cell (i,j) beforeSince the values xij' rijsolution of (1) - (4), we call them the pÆeA&t vaZu.eJ> related to variable3tIt will be shown that in an optimal solution to (1) - (4) most ofXij‘ these ^j 7s will assume one of the preset values.Problem (l)-(4) is a nonlinear programming problem which can beMatuszewski et al. [ 9 ] proposedreduced to the LP format in different ways.We introduce here a slightly different transformation provi­ding more insight into the structure of optimum solutions to problem (l)-(4).such a reduction.Consider the following change of variables, t t ^ tXij = yij + Zij (5)if r*. < twith (i) thenij Xij "ijst ^ t <^-rij < yij < Xij (6a)and0 ^ z!” t (7a)Zij ^ Sij “ Xij ;(ii) if r. t thenij Sij Xijttt(6b)andsrij yij -ijj = 0 ; (7b)and Zij(iii) if < rjj < s^ thent(6c)andyij = "ijt ^ tt (7c);rij “ Xij Zij ij XijAand define the LP problem:2 ((x_ - yjj) + zij) / x±i (8)minimize , t t .(yij,Zij)subject to 2 yC + zC it (9)ij Uj ’y t t? y«+ z- t (10)= w.,ij(ID(6a) - (7c).Clearly the constraints (9) - (11) are equivalent to (2) - (4) andproblem (1) - (4) is feasible if and only if problem (8) - (11) is fea­sible., z^) to (8) - (11) , weWe claim that in an optimum solution (y^and z^. > 0.cannot have both y.. <ijS to y^ and subtracting it from z^For otherwise adding a constantXij ij for the same cell (i,j) violating this condition defines another feasible solution whenever;*-t0 < 5 < Min with objective function value (8) smallerij ‘ "ij5by 2 6/xij *Hence under conditions (9) - (11), minimizing (8) is equivalent to: 2 | (x - y*,) + z!\ I / xi,j ij ij ij ijwhich in turn is equivalent to minimizing (1).(12)minimize5The LP problem (8) - (11) is a generalization of the classicaltransportation problem of linear programming in which parallel arcs(or cells) and both lower and upper bounds on these arcs are allowed.Since this problem has m + n constraints (in addition to the boundconstraints (11)) and since m + n - 1 of these constraints are linearlyindependent [1 ] , [ 12 ] , it follows from LP theory (ibid.) that whenthe problem is feasible there exists an optimum (basic) feasible solu-- 1 variables y^. and z^.ij ijdistinct from their bounds. Thus there is an optimum solution to pro­blem (1) - (4) in which at most m + n - 1 variables x!\ distinct from the pfieA&t v&luZA , r^ and sCtion in which at most m + n assume a valueassume a valueij "In the experiment reported in [ 2 ] , the size of the table was/m = n = 50 and there were 1899 nonzero x s. From the observationsijmade in the previous paragraph it can be expected that at least 1800 coefficients x^ would be equal to some of their preset values, in this study [ 2 ] , the authors solved problem (1) - (4) through a linear programming formulation LP(1) distinct from (8) - (11). They made inte-Indeedresting comparisons between the computed values and the actual 1967 coeffi­cients and they also gave in their Table 2 some details on the numbers ofcomputed values which deviated from the base year coefficients (correspon-- (4)).in (1)ding to x From these data, we infer that 1801 coefficientsijdid not change from their initial value. Furthermore among the remaining98 changing coefficients there may be still some which hit their lowerThe authors also used another formulation LP(2) inor upper bounds.6which all = 0 and s^. = + 00. The corresponding total number of compu-ijted coefficients keeping their base year value was then 1790, that isagain more than 94% of the nonzero cells.The major drawback of the LP approach to this problem seems to bethat it implies that too many coefficients remain to their initial valueswhile it allows quite large variations for the others. Indeed this isthe reason for the introduction of lower and upper bounds in (4), "admit­tedly very arbitrary" [ 9 p. 206] . In addition these methods do not seemappropriate for incorporating additionnai information limiting the possiblevariation of some coefficients: indeed most of the coefficients will eitherkeep the initial value or assume an extreme value. The same observationalso applies for a convex piecewise linear objective function in (1). Inthat case most of the coefficients will assume a value corresponding to abreakpoint of the objective fonction, that is, again a quite arbitraryvalue.In conclusion it seems advisable, on the grounds of these theoreticalremarks, to use a more smooth convex objective function instead of (1).The procedures of Almon, Friedlander and the classical RAS method are of this type - see [8 p.3] for a survey.itselfAccording to Lecomber,"available evidence ( ) suggests that, provided the initial discrepan-t • •cies are not too large, the results are not very sensitive to the methodused. The comparative simplicity of RAS over the other methods that7preserve signs is an overriding asset", [8 p.2]. In addition such methodscan easily take advantage of additional information obtained through complementary methods such as sampling and interview methods [10], [ 3 ] or the interesting proposal recently made by Stemmeulen [ 13] , of usingmethods of factorial analysis.ji8REFERENCES[1 ] PrincetonLinear Programming and Extensions. University Press, Princeton, N.J. (1963).DANTZIG, G.B."A Comparisonof Alternative Methods of Updating Input-Output Coef­ficients". Technological Forecasting and Social Change 10, 79-87 (1977).[2 ] DAVIS, H.C., LOFTING, E.M. and SATHAYE, J.A.[3 ] FONTELA, E., DUVAL, A., GABUS, A., BORLIN, M. and VELAY, C."Forecasting Technical Coefficients and Changes in rela­tive prices", in A.P. Carter and A. Brody (eds.). Appli­cation of Input-Output Analysis. North-Holland, Amsterdam(1970).GIRARD, A. "Etude de méthode de prédiction du trafic téléphonique."INRS Technical Report 78-04, INRS Télécommunications, Verdun, Québec, Canada (1978).[4 ][5 ] Principles of Urban Transport Systems Planning.Washington D.C. (1974).HUTCHINSON, B.G.Scripta Book Co.[6 ] IRELAND, C.T. and KULLBACK, S. "Contingency Tables with Given Marginals." Biometrika 55, 179-188 (1968).KRUITHOF, J. "Telefoonverkeerscrekening." De Ingénieur 52 (8) 15-25 (1937). ~ . . . . . .[7 ][8 ] LECOMBER, J.R.C. "A Critique of Methods of Adjusting, Updating and Projecting Matrices," in R.I.G. Allen and W.F. Goss- ling (eds.). Estimating and Projecting Input-Output Coefficients, Input-Output Publishing Co., London (1975)."Linear Program- Cana-[9 ] MATUSZEWSKI, T.I., PITTS, P.R. and SAWYER, J.A.ming Estimates of Changes in Input Coefficients". _ _dian Journal of Economics and Political Science 30(2),203-210 (1964).9[ 10 ] MIERNYK, W.H. "Sampling techniques in making regional industry forecasts," in A.P. Carter and A. Brody (eds.)» Applica­tion of Input-Output Analysis. North-Holland, Amsterdam(1970).[11 ] NIVERT, K, and von SCHANTZ, C. "Some Applications of Quadratic Programming to the Calculation of Traffic Matrices." Proc. 8th International Teletraffic Congress, pp. 333-1to 333-5 (1976).[12 ] SIMONNARD, M. Programmation linéaire, vol. 1 and 2. Paris, Dunod, (1972).[13 ] STEMMEULEN, E. "Tableaux d'échanges: description et prévision."Cahiers du Bureau Universitaire de Recherche Opérationnelle,Série Recherche, no. 28, 1-83 (1977).[14 ] STEPHAN, F.F. "An Iterative Method of Adjusting Sample Frequency Tables when Expected Marginal Totals Are Known." Annals of Mathematical Statistics 13, 168-178 (1942).gr\■UE DE MONTRÉAL-ÉCOLE POLYTECHNII4919334.fl>\I

Comparison of alternative methods of updating input-output tables : a note

https://publications.polymtl.ca/6002/1/EP-R-79-21_Leblanc.pdf

Comparison of alternative methods of updating input-output tables : a note

Abstract

Similar works

Full text

Available Versions

PolyPublie