Improvement and its computer implementation of an artificial-free simplex-type algorithm by Arsham

Arsham ever described a new phase 1 simplex-type algorithm which allegedly obviates the use of artificial variables. Soon after, Enge and Huhn gave a counterexample, in which Arsham’s algorithm declares the infeasibility of a feasible problem. For this reason, this paper proposes an improvement of Arsham’s algorithm to make it work well, which is searching for the nonbasic variables into the basic variable set (BVS) column by column in only one pivot sequence from beginning to end in order to decrease computational time spent by many repeated searching sequences after each iteration. Next, when the BVS is complete, the phase 1 algorithm ends with a feasible basic solution; otherwise, two variants are presented to pivot the artificial variables out of the basis. The idea of variant 1 is making all artificial variables in basis minimized in turn while maintaining the BVS feasible in the pivoting process. In variant 2, the objective is to minimize the sum of all artificial variables in basis while keeping basic variables (including artificial) feasible. Finally, a computer implementation is accomplished to test the efficiency of our improvement comparing to the ordinary simplex algorithm on some standard test instances and randomly generated problems. The computational results show that our improved algorithms averagely spends much less executive time at each iteration than the ordinary simplex algorithm on sparse linear programming problems, and variant 2 is also competitive on dense linear programming problems.