Simplex tableau

Thus the coordinate transformations (1.18) apply also to the indicator variables and to the objective function. Dh the basis of this observation it is convenient to perform all calculations on a matrix extended by the Zj-cj values and the objective function value as its last row. This extended matrix is the so-called simplex tableau. 

If the j-th column is in the basis then zj cj = 0 follows, but an entry of the last row of the simplex tableau may vanish also for a column that is not in the basis. If this situation occures in the optimal simplex tableau then the linear programming problem has several optimal basic solutions. In our preliminary example this may happen when contour lines of the objective function are parallel to a segment of the boundary of the feasible region. 

NUMBER OF RONS IN FINAL SIMPLEX TABLEAU, N=NE+1 NUMBER OF COLUMNS IN FINAL SIMPLEX TABLEAU,M=NV+LE+GE+1 FINAL SIMPLEX TABLEAU... 

Since the complementary slackness principle does not allow the use of v2 and v3 in the basis, two artificial variables v 2 and v 3 must be introduced. With this basis, x2, x3, u2, vu v 2, and v 3, the tableau of equations is as given below. Notice that the first three lines come directly from the optimal table of the preceding example. The only changes needed in the second trio of equations are in the x columns. The equation below the double line, which is the sum of the fifth and sixth equations, is simply the expression of the sum of artificial variables v 2 + i/3). We shall use the simplex method to minimize this sum, hopefully until it becomes zero. 

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