Sensitivity Analysis in Linear and Nonlinear Programming Models
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Date
2015-07-07
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Addis Ababa University
Abstract
Sensitivity analysis shows how the optimal solution and the value of its ob-
jective function change, given changes in various inputs to the problem. The
task of sensitivity analysis is to _nd out the change limits of data, in other
words the stable interval, so that optimal solution or optimal basis will remain
within its range of optimality. The solution of the optimization problem is
terminated when it reaches a minimum value subject to the given constraints.
If the solution belongs to the boundary of some constraint, such constraints
are called active and the impact of these active constraints on the solution
could be found by acquiring the Lagrange multiplier associated with these
constraints. Since the solution is optimal,the Lagrange multipliers associ-
ated with inequality constraints have nonnegative values [7]. To understand
how sensitive is the optimal solution with respect to the small change in the
right-hand side of the active constraints, the constraints are perturbed and
the new so-obtained optimization problem is solved.
The sensitivity analysis provides information for both linear and nonlin-
ear programming problems, including dual values (in both cases) and range
information (for linear problems only). The dual values for (nonbasic) vari-
ables are called reduced Costs in the case of linear programming problems,
and reduced gradients for nonlinear problems.The dual values for (binding)
constraints are called Shadow Prices for linear programming problems, and
Lagrange Multipliers for nonlinear problems. The active constraints are not
assumed to remain active if the problem data are perturbed, nor the partial
derivatives are assumed to exist. In other words, all the elements, variables,
parameters, KKT multipliers, and objective function values may vary pro-
vided that optimality is maintained and the general structure of a feasible
perturbation.
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Keywords
Sensitivity, Analysis in Linear, Nonlinear, Programming Models