6.3 New variables and new constraints

A new variable \(x_{n + 1}\) with column \(a\) and profit \(c_{n + 1}\) is priced out with the existing shadow prices: its reduced profit is \(c_{n + 1} - y^{\mathsf T}a\). If this is \(\leq 0\) the present plan stays optimal and the new activity is not worth starting; if it is positive, \(x_{n + 1}\) enters the basis and the primal simplex method continues from the old optimum. A product using one unit of each resource (\(a = (1, 1, 1)\)) is worth making in the running example only if its profit exceeds \(y^{\mathsf T}a = \frac 53 + \frac {10}3 + 0 = 5\), the value of the resources it would consume (Software 6.5.3).

A new constraint that the present optimum satisfies changes nothing. If it is violated, add it with its slack, express it in the current basis, and the tableau is dual feasible but primal infeasible: the dual simplex method of the section on duality restores optimality, usually in a few pivots.

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.