The Solution of Simultaneous Equations

Abstract

Overhead distribution involves the transfer of costs between where T, represents the total costs for which we are solving, the overhead and production departments. This cost acI, are the initial direct costs, and a, represents the percentage counting problem is common to many businesses. As demof the total cost of department “i” which is part of the total onstrated by the IBM Type 650 Magnetic Drum Data Proccost of department “j”. essing Machine, this sample application requires the soluThe method of solution is that known to mathematition of ninety algebraic linear equations. cians as the Gauss-Seidel Method. It is an iterative proceHere (in this problem recently solved by a Philadelphia dure whereby approximate solutions are improved by rebank) total costs are determined for ninety departments peated calculations until the correct solutions are obtained. where there are six hundred and twenty-five interdepartThe correct solution is automatically determined by the mental distributions. Mathematically this is expressed as calculator and the iteration is ended when the sum of the follows: differences between the total costs on one iteration and the total costs on the previous iteration is less than one dollar. Following the solution a card for each department is punched T1 = I, + a2,1Tz + a3,1T3 + --+ a90,1T9D showing department number, total cost, distributed amount, and net cost. TS = I, + al,zT1 + a3,2T3 + --+ ago,zT90 Two hundred and ten instructions plus a few constants ______-------------~~~~~~~~~~---are loaded in less than one minute. Nine iterations are required at a process rate of forty seconds per iteration. The ______-----------_---~~~~~~~~~~-ninety answer cards are punched in less than a minute. The ____--------------~______I_______ total running time from the first card fed to the last solution