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Right Hand Side Changes


For these types of changes, we concentrate on maximization problems with all tex2html_wrap_inline332 constraints. Other cases are handled similarly.

Take the following problem:


The optimal tableau, after adding slacks tex2html_wrap_inline286 and tex2html_wrap_inline288 is


Now suppose instead of 12 units in the first constraint, we only had 11. This is equivalent to forcing tex2html_wrap_inline286 to take on value 1. Writing the constraints in the optimal tableau long-hand, we get




If we force tex2html_wrap_inline286 to 1 and keep tex2html_wrap_inline288 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 tex2html_wrap_inline358 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 tex2html_wrap_inline280 between -2 and 3 in order for the basis not to change. For any tex2html_wrap_inline280 in that range, the optimal objective will be tex2html_wrap_inline368 . For example, with tex2html_wrap_inline280 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 tex2html_wrap_inline378 in the original formulation, we get an objective of tex2html_wrap_inline380 in the final tableau, as long as tex2html_wrap_inline310 .

Perhaps the most important concept in sensitivity analysis is the shadow price tex2html_wrap_inline384 of a constraint: If the RHS of Constraint i changes by tex2html_wrap_inline280 in the original formulation, the optimal objective value changes by tex2html_wrap_inline388 . The shadow price tex2html_wrap_inline384 can be found in the optimal tableau. It is the reduced cost of the slack variable tex2html_wrap_inline392 . So it is found in the cost row (Row 0) in the column corresponding the slack for Constraint i. In this case, tex2html_wrap_inline394 (found in Row 0 in the column of tex2html_wrap_inline286 ) and tex2html_wrap_inline398 (found in Row 0 in the column of tex2html_wrap_inline288 ). The value tex2html_wrap_inline384 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 tex2html_wrap_inline404 . 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.

next up previous contents
Next: New Variable Up: Tableau Sensitivity Analysis Previous: Cost Changes

Michael A. Trick
Mon Aug 24 15:42:04 EDT 1998