For these types of changes, we concentrate on maximization problems with all constraints. Other cases are handled similarly.
Take the following problem:
The optimal tableau, after adding slacks and is
Now suppose instead of 12 units in the first constraint, we only had 11. This is equivalent to forcing to take on value 1. Writing the constraints in the optimal tableau long-hand, we get
If we force to 1 and keep at zero (as a nonbasic variable should be), the new solution would be z = 21, y=1, x=4. Since all variables are nonnegative, this is the optimal solution.
In general, changing the amount of the right-hand-side from 12 to in the first constraint changes the tableau to:
This represents an optimal tableau as long as the righthand side is all non-negative. In other words, we need between -2 and 3 in order for the basis not to change. For any in that range, the optimal objective will be . For example, with equals 2, the new objective is 24 with y=4 and x=1.
Similarly, if we change the right-hand-side of the second constraint from 5 to in the original formulation, we get an objective of in the final tableau, as long as .
Perhaps the most important concept in sensitivity analysis is the shadow price of a constraint: If the RHS of Constraint i changes by in the original formulation, the optimal objective value changes by . The shadow price can be found in the optimal tableau. It is the reduced cost of the slack variable . So it is found in the cost row (Row 0) in the column corresponding the slack for Constraint i. In this case, (found in Row 0 in the column of ) and (found in Row 0 in the column of ). The value is really the marginal value of the resource associated with Constraint i. For example, the optimal objective value (currently 22) would increase by 2 if we could increase the RHS of the second constraint by . In other words, the marginal value of that resource is 2, i.e. we are willing to pay up to 2 to increase the right hand side of the second constraint by 1 unit. You may have noticed the similarity of interpretation between shadow prices in linear programming and Lagrange multipliers in constrained optimization. Is this just a coincidence? Of course not. This parallel should not be too surprising since, after all, linear programming is a special case of constrained optimization. To derive this equivalence (between shadow prices and optimal Lagrange multipiers), one could write the KKT conditions for the linear program...but we will skip this in this course!
In summary, changing the right-hand-side of a constraint is identical to setting the corresponding slack variable to some value. This gives us the shadow price (which equals the reduced cost for the corresponding slack) and the ranges.