Let us now return to return to the problem:
We will now consider the meaning of the
First we shall address Objective Ranging. Consider our current objective function:
Figure 7 shows this objective function as the line PQ. It touches the feasible region OABC at the vertex B which is therefore the optimal solution.
Let us now consider the cost coefficient of variable x _{1} in the objective function. If we reduce the value of x_{1}'s cost coefficient from - 5 to - 6 the new objective function:
is represented by the line P'Q'. Note that the optimal solution to the problem still lies at vertex B. Now let us decrease x
is represented by line P*Q* shown in Figure 8.
Note that there is now a choice of optimal solutions at any point along the boundary AB of the feasible region. Because we consider solutions at vertices only we have a choice of solutions at vertex B or A. Suppose we now decrease x _{1}'s cost coefficient just beyond - 8. The
objective function P*Q* will get slightly steeper and the optimal
solution will lie at vertex A only. For instance, decreasing x_{1}'s cost coefficient to - 10 gives an objective function of:
represented by line P We can therefore deduce that the cost coefficient of variable x The value - 8.0 is sometimes referred to as x By an almost identical argument, a variable's
We will now consider Right Hand Side Ranging. Earlier, we analyzed the effect of relaxing the right hand side of ROW2 by 0.5 in our problem:
Figure 9 shows the relaxed ROW2 constraint as line E'F'. The new feasible region of the problem becomes OA'B'C.
Suppose we now consider the effect of relaxing ROW2 by a further 1.3 to 6.8. ROW2 now appears as:
Figure 10 shows the geometric representation of the modified problem.
ROW2 is represented by the line E*F*. The feasible region is now ODC. The objective function is represented by the dotted line and shows the optimal solution to lie at vertex D where:
Note from our definition of shadow prices that the objective function at vertex D is given by:
i.e.
Suppose we now relax ROW2 further than 1.8, say by a further 1.2 to 8.0 so ROW2 becomes:
The line E We can therefore deduce that ROW2 can be relaxed from its original RHS value of 5.0 to 6.8 and cause the objective function value to decrease by ROW2's shadow price per unit relaxation of the right hand side. Relaxing ROW2 beyond 6.8 has no effect on the optimal solution to the problem. The value 6.8 is sometimes referred to as ROW2's By an almost identical argument, a row's Note
from figure 10 that if we tighten ROW2 from a RHS of 5.0 the feasible
region will slowly be reduced until it is represented by OAC. This
occurs when the RHS of ROW2 is reduced to 4.0. The optimal solution to
the problem will now lie at point C. Tightening ROW2 further Since
the upper and lower activities for each row are concerned with how much
the constraint can be relaxed or tightened to give a |

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