What is the amount that the objective function coefficient of a decision variable would have to improve before that variable would have a positive value in the solution?

Ignore?

Sensitivity Analysis (a/k/a post-optimality analysis)

Examines how the optimal solution is affected by changes, within specified ranges, in: the objective function coefficient; the RHS values.

Range of optimality for each coefficient provides the range of values over which the current solution will remain optimal. Graphically, limits of a range of optimality are found by changing the slope of the objec…

The change in the value of the optimal solution per unit increase in the RHS. Dual value is equal to the difference in the values of the objective functions between new and original problems. …

Reduced cost or a decision variable whose value is 0 in optimal solution is amount variable objective function coefficient would have to improve (increase for maximization problems, decrease for minimization problems) before variable could assume positive value.

The range of the RHS constraints over which dual value is applicable. (Range of feasibility measures RHS values for which dual prices will not change).

Resources that depend on amount of resources used by decision variables. Relevant costs are reflected in objective function coefficients.

Resource cost that must be paid regardless of amount of resource actually used by decision variables. Sunk costs are not reflected in objective function coefficients.

Limitations of Classical Computer Solutions of LP

Sensitivity analysis information in computer output is based on assumption of 1 coefficient change. When using Solver, range analysis for objective function coefficients and constraint RHS can NOT be done for simultaneous changes for mult…

Media selection problems usually determine

how many times to use each media source.

Let M be the number of units to make and B the number of units to buy. If it costs $2 to make a unit and $3 to buy a unit and 4000 units…

P12+P22+P32 is less than or equal to 400

If Pij = the production of product i in period j, then to indicate that the limit on production of the company's three products in period 2 is 400,

P13-P14 is less than or equal to 100; P14-P13 is less than or equal to 100.

Let Pij = the production of product i in period j. To specify that production of product 1 in period 3 and in period 4 differs by no more than 100 units `

return from additional investment funds

For a portfolio selection problem with the objective of maximizing expected return, the dual price for the available funds constraint provides information about the

if more funds can be obtained at a rate of 5.5%, some should be.

The objective of an investment is to maximize return. If dual price for a constraint that compares funds used with funds available is .058, this means that

2x1+3x2 is less than or equal to 100.

A company makes two products from steel; one requires 2 tons of steel and the other requires 3 tons. There are 100 tons of steel available daily. A constraint on daily production could be written as:

private firms usually find it profitable to produce public goods? T/F

Using minutes as the unit of measurement on the left-hand side of a constraint and using hours on the RHS is acceptable since both are a measure of time.

How should the 2M be allocated to each of the loan/investment alternatives to maximize total annual return? What is the projected total annual return?

when solving an LP problem we assume that values of all model coefficients are known with certainty.

Approaches to Sensitivity Analysis

-Change the data and re-solve the model!

Solver's Sensitivity Report Answers Questions About:

amounts by which objective function coefficients can change without changing the optimal solution.

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.

-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.

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

Changing a RHS Value for a Binding Constraint

Also changes the feasible region and the optimal solution

You must re-solve the problem

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

The Shadow Prices of Resources

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

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.

Will not be produced in an optimal solution.

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

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

Two Cases Can Occur with Simultaneous Changes in Objective Function Coefficients

-Case 1: All variables with changed obj. coefficients have nonzero reduced costs.

Simultaneous Changes in Objective Function Coefficients: Case 1

(All variables with changed obj. coefficients have nonzero reduced costs.)

Simultaneous Changes in Objective Function Coefficients: Case 2

(At least one variable with changed obj. coefficient has a reduced cost of zero.)

A Warning About Degeneracy

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

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

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

Returns the current optimization # (O#)

Returns the current parameter # (P#)

Returns the current iteration #

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

To return the values for an output celll across multiple optimizations

traditional sensitivity analysis assumes all coefficients in a model are known with certainty