two or more constraints contradict each other ("mathematical incon- 

 sistency") . 



When either of these situations occur following this technique, defi- 

 nite and unique events will stop the iteration process. If a maximum 

 objective is called for when no upper limit is set hy the constraints, the 

 matrix-iteration process will run into a dead end hy having a positive 

 value in the l)ottom row. indicating that the solution can he made larger, 

 and positive values in tlie last column, but there will be no negative 

 elements to qualify as a pivot element (negative Q-values cannot he gen- 

 erated I . 



If two or more constraints contradict each other, the pivoting pro- 

 cess will dead-end hecause a negative value will appear in the last col- 

 umn, but there will be no positive element to qualify as a pivot element 

 (again, no negative Q-values can he generated). 



Thus, tests for "l>oundedness" and mathematical consistency are 

 built into this technique. In the accompanying computer program (Ap- 

 pendix A), error messages to this effect are included. 



Computer Program 



A Fortran II program for this L. P. problem solving technique is 

 presented in Appendix A. 



It is arljitrarily dimensioned for a 14 x 14 matrix which wovild be 

 the maximum size prol)leni that could be run in a 20,000 unit storage 

 capacity computer if numerical tables and/or other management rou- 

 tines have reduced the storage area to approximately 8.000 units. 



The output gives the initial matrix, the activities being pivoted and 

 the objective function for each iteration, the final solution activity 

 levels, the final objective function (solution) and the shadow prices. 



Selected References 



(1) Bauniol, William J., Economic Theory and Operations Analysis. 

 (Englewood Cliffs; Prentice-Hall Inc., 1961). 



(2) Heady, Earl O. and Wilfred Candler, Linear Programming Methods. 

 (Ames: The Iowa State University Press, 1963*. 



(3) Stiefel, Eduard L., An Introduction to Numerical Mathematics. 

 (New York: Academic Press, 1963). 



(4) Vajda, S., An Introduction to Linear Programming and the Theory of Games. 

 (New York: John Wiley & Sons, Inc., 1960). 



(5) Vajda, S., Mathematical Programming. 



(Reading, Mass.: Addison-Wesley Publishing Company, Inc., 1961). 



13 



