Chapter 4 Sensitivity Analysis

Reduced Cost

For each product equals its per-unit marginal profit minus the per-unit value of the resources it consumes (priced at their shadow prices)

Resources in Excess Supply

Have a shadow price (or marginal value) of zero.

The Reduced Cost of a Product

Is the difference between its marginal profit and the marginal value of the resources it consumes.

O# - v *(P# -1)

Returns the current iteration #

PsiCurrentOpt( )

Returns the current optimization # (O#)

INT( (O# -1)/v )+1

Returns the current parameter # (P#)

Solver Tables

Summarize the optimal value of multiple output cells as changes are made to a single input cell.

A Warning About Degeneracy

The solution to an LP problem is degenerate if the Allowable Increase or Decrease on any constraint is zero (0).

Use PsiOptValue( )

To return the values for an output celll across multiple optimizations

Use PsiOptParam( )

To specify values for an input cell to take on across multiple optimizations

Changes in Objective Function Coefficients

Values in the "Allowable Increase" and "Allowable Decrease" columns for the Changing Cells indicate the amounts by which an objective function coefficient can change without changing the optimal solution, assuming all other coefficients remain constant.

Alternate Optimal Solutions

Values of zero (0) in the "Allowable Increase" or "Allowable Decrease" columns for the Changing Cells indicate that an alternate optimal solution exists.

Products whose marginal profits are less than the marginal value of the goods required for their production

Will not be produced in an optimal solution.

To Find the Optimal Solution After Changing a Binding RHS Value,

You must re-solve the problem

Simultaneous Changes in Objective Function Coefficients: Case 1

(All variables with changed obj. coefficients have nonzero reduced costs.) -The current solution remains optimal provided the obj. coefficient changes are all within their Allowable Increase or Decrease.

Simultaneous Changes in Objective Function Coefficients: Case 2

(At least one variable with changed obj. coefficient has a reduced cost of zero.) -If more than one objective function coefficient changes, the current solution remains optimal provided the rj sum to <= 1. -If the rj sum to > 1, the current solution, might remain optimal, but this is not guaranteed.

Solver's Sensitivity Report Answers Questions About:

-Amounts by which objective function coefficients can change without changing the optimal solution. -The impact on the optimal objective function value of changes in constrained resources. -The impact on the optimal objective function value of forced changes in decision variables. -The impact changes in constraint coefficients will have on the optimal solution.

Two Cases Can Occur with Simultaneous Changes in Objective Function Coefficients

-Case 1: All variables with changed obj. coefficients have nonzero reduced costs. -Case 2: At least one variable with changed obj. coefficient has a reduced cost of zero.

Approaches to Sensitivity Analysis

-Change the data and re-solve the model! ~Sometimes this is the only practical approach. -Solver also produces sensitivity reports that can answer various question.

Shadow Price

-Of a constraint indicates the amount by which the objective function value changes given a unit increase in the RHS value of the constraint, assuming all other coefficients remain constant. -Hold only within RHS changes falling within the values in "Allowable Increase" and "Allowable Decrease" columns. -For nonbinding constraints are always zero.

Shadow Prices

-Only indicate the changes that occur in the objective function value as RHS values change.

We can use RSP's ability to run multiple parameterized optimizations to carry out ad hoc sensitivity such as:

-Spider Tables & Plots -Solver Tables

Spider Tables & Plots

-Summarize the optimal value for one output cell as individual changes are made to various input cells. -To vary each of p parameters by v values requires a total of p*v optimizations

Robust Optimization

-Traditional sensitivity analysis assumes all coefficients in a model are known with certainty -Some argue this makes optimal solutions on the boundaries of feasible regions very "fragile" -A "robust" solution to an LP occurs in the interior of the feasible region and remains feasible and reasonably good for modest changes in model coefficients

Sensitivity Analysis

-When solving an LP problem we assume that values of all model coefficients are known with certainty. -Such certainty rarely exists. -Helps answer questions about how sensitive the optimal solution is to changes in various coefficients in a model.

When the Solution is Degenerate:

1.The methods mentioned earlier for detecting alternate optimal solutions cannot be relied upon. 2.The reduced costs for the changing cells may not be unique. Also, the objective function coefficients for changing cells must change by at least as much as (and possibly more than) their respective reduced costs before the optimal solution would change. 3. The allowable increases and decreases for the objective function coefficients still hold and, in fact, the coefficients may have to be changed beyond the allowable increase and decrease limits before the optimal solution changes. 4. The given shadow prices and their ranges may still be interpreted in the usual way but they may not be unique. That is, a different set of shadow prices and ranges may also apply to the problem (even if the optimal solution is unique).

Changing a RHS Value for a Binding Constraint

Also changes the feasible region and the optimal solution

The 100% Rule

Can be used to determine if the optimal solutions changes when more than one objective function coefficient changes.

The Shadow Prices of Resources

Equate the marginal value of the resources consumed with the marginal benefit of the goods being produced.