Origin not feasible
The problems of which all bi are positive are made with Realizable Origin. It's easy to have a basic fix and the simplex is compatible. For the problems at the origin not realizable, one initially seeks to solve the Auxiliary Problem.
In the auxiliary problem, we add an auxiliary variable x0. This variable is included in all the constraints. We seek to minimize its value (maximize its opposite).
The first iteration is specific, we force the auxiliary variable to enter. The pivot line is that of which the bi is the smallest. The following follows the classical resolution of a simplex.
Once the simplex is optimal, we express z as a function of the non-base variables. The origin of the base variables is then achievable (here the blue boxes show the evolution of the stresses by the resolution of the simplex).
The new problem to be solved is as follows:
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