In this generalized matrix, Sp is our eventual pivot row and X q 

 is our eventual pivot column (i.e. a p, is our eventual pivot element). 

 There are two requirements on which we will insist: 



A. Only solutions that are acceptahle values will be considered 



(Xi ^0, X2 ^0, ...X„ ^0, 81^0,82^0 S^^O), 



therefore, the C's (last column values) must remain or become 



B. During each exchange, C* (the objective solution) must in- 

 crease (at least not decrease). 



IF ALL C'S ARE NON-NEGATIVE 



From the above requirements and the pivoting rules (assuming all 

 C.s are non-negative), three conclusions can be drawn: 



1-1. 8ince Cp -^ (becomes) — C,, /pivot (pivoting rule #3), 

 which must remain ^s:0, we must choose a pivot element 

 that is negative. 



1-2. 8ince C* — » C* — (Cpaq)/pivot (pivoting rule 4^4), which 

 must be non-decreasing, we must choose a pivot column so 

 that the quantity (CpEq ) /pivot is — 0, i.e. aq must be posi- 

 tive. 



1-3. Since Ck — > C ^ — (Cpakq ) /pivot (pivoting rule 1^4:) which 

 must remain :^0, we must choose a pivot row so that C,^^^ 

 (Cj. aikq ) /pivot. 



Conclusion 1-3 is automatically satisfied if a^^^ ^^ 0. 



However, if there is more than one a jq in the pivot column which 

 is negative, we will gain by being more cautious. 



Suppose a ^ is negative, then Conclusion 1-3 can be written: 



Ck /a k,j ^ C p /pivot, or C k /a kq ^Cp/a 



pq 



If we call C^/an,,| , "Qn," (the kth "characteristic quotient"), then the 

 gain comes by choosing the pivot row so that Q ], is the largest of the Q's, 

 for this will prevent any of the positive C's from becoming negative. 

 Since the Q's in which we are interested are all negative, we can con- 

 clude that our pivot row is the one which has the smallest absolute 

 value of Q. 



From these three conclusions, three rules can be made for deter- 

 mining the appropriate pivot row and pivot column for the pivoting 

 process if all C's are non-negative. 



22 



