Equality Constraints and No Solution 



Situations 



There are two items that warrant hrief coverage due to their special 

 handling in the specific technique described herein. 



The Equality Constraint 



Because this technique does not use artificial varial)les as do the 

 more conventional methods, it is recommended tliat the equality con- 

 straint he handled as two opposite inequality constraints, adding and 

 subtracting slack varialiles wliere indicated. Handled in this manner, 

 one of the two slack variables will equal zero in tlie final solution. 



It is also possible to use one of the unknowns as if it were a slack 

 variable. In the constraint X^ -f X^ = 100, for example. X, or X^ 

 could be used to absorb the "unused" portion of the 100, thus developing 

 the constraint: Xj = — X2 + 100. 



Now that Xi has been "solved" (in terms of Xo ) , however, it be- 

 comes necessary to restate all other constraints (that contain X, ), and 

 the objective function, in terms of X2. 



In our example problem, the constraint 2Xi + 8X5 — 24, becomes 

 2( — Xo -\- 100) -|- 3X2 — 24, and the representative equality becomes: 

 Sj =r — X2 — 176. The objective function. Z = SX^ -(- 2X0 max.) be- 

 comes X = 3(— X2 + 100) + 2X2, or Z = — Xo + 300. and the ini- 

 tial solution is no longer zero but 300. 



An equality constraint handled in this manner reduces the number 

 of constraints by one, and reduces the number of iterations by two, but, 

 as is quite evident, requires a great deal more time in setting up the 

 problem. 



This increased "set up" time and the increased probability of mak- 

 ing simple mathematical errors while restating the constraints and ob- 

 jective function, appear to the authors as justification for handling an 

 equality constraint in the first described manner. Therefore, unless the 

 cost of the additional "solving time" is prohibitive, or the additional re- 

 quired constraint causes the problem to become too unmanageable, it is 

 recommended that an equality constraint be handled as two opposite 

 inequality constraints. 



The "No Solution" Situation 



The two most common "no solution" situations that occur in an 

 L. P. problem are: (1) when a function is to be maximized, but there 

 is no upper-limit constraint ("unljounded solution"), and (2) when 



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