8 The two-phase method.
8.1 The two-phase method.
The simplex algorithm assumes that the initial point is feasible in the primal problem. If b is
greater than or equal to zero, then the origin is feasible. If the origin is not feasible, then it is
necessary to determine some other initial point that is feasible. It is possible to introduce an
auxiliary primal problem specifically designed to help in this task. The first phase of the two-
phase method applies the simplex algorithm to the auxiliary problem. The second phase uses the
feasible point generated in the first phase as initial point in the original problem. There is
typically a need for elementary row operations to bring the tableau into the form required by the
simplex algorithm.
8.2 The auxiliary problem.
Consider the primal problem
12
12 12
12
22
max 9 8x when 3 3
,0
xx
xxx
xx
−− ≤−
−− −−≤−
≥
.
The feasible set does not include the origin. The augmented system has the two equations
121
122
22
33
xxy
xxy
−− +=−
−−+=−.
Write the system in the form
1 211
1222
22
33
xxyz
xxyz
+−+=
+−+=
,
using the auxiliary non-negative variables 12
,
zz. The auxiliary problem seeks to minimize 12
zz+
without violating the previous system.
8.3 Tableaux.
If the minimizer is given by 12
0zz==, then the previous system yields a solution that is
feasible in the original problem. The tableau associated with the problem to maximize 12
zz−− is
given by
12 10102
310 1013
0000 110
−
−
−−
.
Observe how there are no basic variables. This is readily fixed by adding the first and the second
row to the last row. It is convenient to extend the tableau with one more row for this particular
