The Technique for Solution 



In the following explanation each figure in the matrix will he called 

 an "element." 



The problem is solved hy attempting to bring Xj and X2 (the ac- 

 tivities) into solution. The resources (Sj ) will be used to produce the 

 activities (Xj). This will be done by mathematically exchanging rows 

 and columns of the matrix, a row and a column at a time, in a series 

 of matrix-forming processes called "iterations." 



The process of exchanging a row and a column is called "pivoting" 

 and is accomplished by using a set of rules described on page 4 and 

 derived in Appendix B. The row and column that are exchanged in a 

 pivoting process are known as "pivot row" and "pivot column," and the 

 common element is known as the "pivot element." Under no circum- 

 stance will the bottom row or last column be considered as the pivot row 

 or pivot column. 



Pivot Column. With the exception of the last column, any column 

 with a positive bottom element can be used as the pivot column,^ as 

 long as it contains a potential pivot element (see below). -^ The pro- 

 cedure outlined in this l)ulletin specifies the pivot column as the one 

 with the largest positive bottom element, but actually the choice is ar- 

 bitrary. In the example proljlem the pivot column will be the X^ 

 column. 



Pivot Row. The appropriate row or "pivot row" is determined by 

 the resource that is most limiting in the production of the pivot column 

 activity. This of course excludes the bottom or objective function row. 

 According to the constraints of the example problem (and also the 

 matrix table if signs arc ignored), the available supply of S^ (24) 

 will allow 12 units of X^ to be produced, the available supply of S^ 

 (three) will allow 6 units of X^ to he produced, and the available sup- 

 ply of S3 (five) will allow 5 units of X^ to be produced. The limiting 

 resource then is S3, so the S3 row becomes the pivot row. The element 

 at the intersection of the pivot column and pivot row is the "pivot 

 element." This determination of the pivot row can be done quickly by 

 generating what will be called "Q-values." This is done for each poten- 

 tial pivot row by dividing the resource value (in the last column) by the 

 element in the pivot column, and registering this quotient in a "Q- 

 column" to the right of the table. After all Q's have been determined, 

 the row with the smallest absolute value of Q becomes the pivot row. 



1 The algebraic logic of this is also covered in Appendix B. 



2 This is also true if the problem is one of minimization, as will be explained 

 on page 7. 



