Pivot element: The pivot element must be negative. 



Pivot column: The bottom element in the pivot column must be 

 positive. 



Pivot row : Among rows with negative a i,, 's, the pivot row is the 

 row which has the smallest absolute value of (}. 



IF SOME C'S ARE NEGATIVE 



A linear programming problem that contains greater-than-or-equal- 

 to constraints will have negative values appearing in the last column 

 of the initial table. These negative values must become non-negative 

 (see requirement A, page 22). Given this situation and considering 

 the pivoting rules, three conclusions can be drawn: 



2-1. Since we are only concerned with the negative C's, only rows 

 with negative C's will be considered for the pivot row. 



2-2. Since C ,, — ^ — C ,, /pivot, we must choose a positive pivot 

 element (i.e. a ,„, > 0). 



2-3. If there is only one row that contains a negative C and at 

 least one positive element in that row, there is no problem 

 of determining the appropriate pivot row and pivot column 

 for the pivoting process (the choice of a pivot column is ar- 

 batrary). However, if there are two or more rows containing 

 negative C's, we will gain by being cautious. 



Suppose that C ^^ is negative and a i^q is positive (see generalized 

 table page 21) . 



Since C I, -^ C ^ — (C ,, a k,, ) /pivot, it would be desirable if 

 (C|, a kq ) /pivot were ^ C k since this would cause Ck to become 

 non-negative also. This can be modified to read : C p /pivot — C ,^ /a kq 

 or Cp/a^^i i^ Ck/akq or even further Qp— Q^. Since again the Q 

 values we are dealing with are all negative, we can conclude that we 

 would gain if the pivot row is one which has the largest absolute value 

 of Q.i 



From these three conclusions (2-1, 2-2, and 2-3), four rules can be 

 made for determining the appropriate pivot row and column for the 

 pivoting process if some C's are negative. 



^ It is possible that a previously non-negative C may become negative during 

 a pivoting process. This is of no concern, however, since it will eventually be 

 eliminated in the same manner as the other negative C's. 



23 



