During the solution of a production‐planning linear program with three decision variables and two binding resource constraints, the primal simplex method reaches an objective value of 19 500. The algorithm then performs eight further pivot steps in which the basis changes but the objective value and the decision-variable vector remain exactly the same; one slack variable that is basic keeps a value of 0 throughout these stalls. Which boundary-case condition best explains this behaviour, and what is the most appropriate action to guarantee that the simplex procedure eventually terminates?
The model is infeasible; restart with a Phase I artificial-variable formulation to locate a feasible basic solution first.
The problem is unbounded; add an explicit upper-bound constraint on the entering variable so the objective cannot diverge.
Alternate optimal solutions exist along an edge; perturb the objective coefficients with a small ε to force a unique interior optimum.
Degeneracy is causing the stalls; apply an anti-cycling or lexicographic pivot rule (for example, Bland's rule) to guarantee finite convergence.
A basic feasible solution in which at least one basic variable is zero is degenerate. When degeneracy occurs, a pivot can change the basis without changing the solution or the objective, so the simplex method may stall or even cycle indefinitely. Anti-cycling or lexicographic pivot rules such as Bland's rule break ties in a consistent way and prove that the algorithm will finish after a finite number of steps. The existence of alternate optimal (multiple) solutions also yields zero improvement in the objective, but the solution itself changes-there is an entire edge of optimal points rather than repeated stalling at one vertex. Unboundedness would be signalled by the objective moving without limit, not by staying constant, and infeasibility is detected in Phase I, before any optimal objective value is found, not after a feasible solution with value 19 500 exists.
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