ch04 1

Chapter 4

Linear Optimization:

Sensitivity Analysis

Chapter 4

Linear Optimization:

Sensitivity Analysis

DECISION MODELING

DECISION MODELING

WITH

WITH

MICROSOFT EXCEL

MICROSOFT EXCEL

Sensitivity

Analysis

Sensitivity

Analysis

Sensitivity analysis

Sensitivity analysis is the effect a (small) change of an exogenous

variable has upon another variable. In the case of optimization models, sensitivity analysis refers to the process of analyzing such changes in a model after the optimal solution has been found.

We will use 2-dimensional graphs (i.e., only two decision variables) to give insight into LP

sensitivity analysis. In addition, we will use Solver Sensitivity Report and SolverTable.

Oak Products

Model

Oak Products

Model

Let’s return to the Oak Products model of Chapter 3. In this simplified model, there are only two decision variables:

Captains chairs (C ) Mates chairs (M )

Now, review the symbolic model

containing the objective function and objective function constraints

Max 56C + 40M (objective function) Subject to 8C + 4M < 1280 (Long Dowels Restriction) 4C + 12M < 1600 (Short Dowels Restriction) C + M > 100 (Minimum Production) 4C + 4M < 760 (Legs Restriction) C < 140 (Heavy Seats Restriction) M < 120 (Light Seats Restriction) C > 0 and M > 0 (Nonnegativity Conditions)

Symbolic

Model

Symbolic

Model

Using GLP

Using GLP

Graphic LP Optimizer (GLP) to analyze the LP model.

Type labels for the X,Y axes, enter up to 6 constraints, and the objective function (PAYOFF) here.

This area is the graphing pane.

First, obtain a graphical portrayal of the constraint set.

For this example, M is arbitrarily assigned to the vertical axis.

Start with the Long Dowels (LDOWELS) constraint. Note that the “:” separates the label from the constraint. This area represents the feasible region

feasible region (the set of all non-negative points that satisfies the Long Dowels constraint).

Note that you can drag the constraint line and observe what happens to the feasible region.

Only the RHS of the equation will change since the slope is fixed by the ratio of the two coefficients in the

Add the remaining constraints:

Short Dowels

Notice how the feasible region changes as we add

Add the remaining constraints:

Add the remaining constraints:

Add the remaining constraints:

Add the remaining constraints:

Feasible

Region

Feasible

Region

Remember, the Feasible Region is Feasible Region

the region that satisfies all the constraints.

NOTE:

Adding additional constraints can never enlarge the Feasible Region, but leaves it Feasible Region

either unchanged or smaller.

Deleting constraints leaves the Feasible Feasible Region

Optimal

Solution

Optimal

Solution

Therefore, the optimal solutionoptimal solution will be a feasible production alternative (i.e., the number of

Captain and Mate chairs) that gives the highest possible value to the objective function (i.e.,

maximum profit contribution).

A feasible solutionfeasible solution (or feasible decision) is any pair of values for C and M that satisfy all the constraints.

Remember, the values for C and M represent the quantities of each product that will be produced.

After entering the objective function, a dashed line appears in the graph.

This line gives the set of all values of C and M that produce exactly $2000 profit (as specified in the dialog), and is thus called the $2000 isoprofit lineisoprofit line. The slope of this line is fixed, but the axes

intercept will change as you change the profitability from $2000.

In fact, you can view changes in the Profit by dragging the line in the graph.

It would not make sense to drag beyond the boundaries of the feasible region.

To find the optimal solution using GLP, click on the AutoMax button.

Use the Scissors button to clean up

Note that the isoprofit line intersects the feasible region at one point. This point is called an unique unique

optimal solution

optimal solution to our model.

The maximum profit is $9,680, which is the same value we obtained using Solver. These two constraints are binding.

Now that we know an optimal solution exists, just what is the unique optimal solutionunique optimal solution to our model?

The coordinates of this point are displayed in the status bar at the bottom of the program.

This means that we will make 130 Captains chairs and 60 Mates chairs. This is the optimal solution optimal solution

(or just solution).

56(130) + 40(60) = $9,680

The optimal objective valueoptimal objective value (or just optimal

value, OV) is calculated using the above solution

The optimal solution to any LP model will never

occur at an interior point of the feasible region.

Geometrically, a binding constraint is one that passes through the optimal solution.

Geometrically, a non-binding constraint is one that does not pass through the optimal

solution.

Important

Notes

Important

Notes

The following notes apply to all LP solutions:

Adding constraints to a model will either impair the OV or leave it unchanged.

Deleting constraints will either improve the OV

or leave it unchanged.

Adding decision variables will either improve the OV or leave it unchanged.

Deleting decision variables will either impair the

OV or leave it unchanged.

Important

Notes

Important

Notes

The following notes apply to all LP solutions:

Extreme

Points

Extreme

Points

The corners of the feasible region are called Extreme PointsExtreme Points.

The solution to the model will occur at one of the extreme

To illustrate this, in GLP, change only the profit-per-chair contribution margin for the Mate chair from $40 to $80 per unit (i.e., change the

objective function to: 56C + 80M=9680).

Then, press on the AutoMax button to find the optimal solution using this new objective

As you can see, the feasible region remains un-

changed. However, there is now a new optimal point.

Note also, these intersecting constraints are now binding. This optimal point also lies at an extreme (corner) point.

f

g

h

A B

C

Examine this six-sided constraint set and contours of 3 different objective functions (f, g, h).

Note the direction each objective function is pointing. In each case there is an optimal solution at a corner.

f h g Objective Solution C A B & C and the edge connecting these points

f g h A B C

Notice objective function g.

In an LP model, if there is an optimal solution, there is always at least one optimal corner solution.

This occurs when the objective function contour has the same slope as the constraint, resulting in multiple optimal solutions. This is a case of multiple optimamultiple optima (or alternative alternative

optima

optima) when the objective contour coincides with one of the constraint lines on the boundary of the feasible region.

Minimization

Model

Minimization

Model

The graphical method can also be applied to a

minimization (Min) model. The only difference between Max and Min LP models is the optimizing

direction of the objective function contours.

When maximizing, the

objective function moves “uphill,”

while minimizing moves the objective function “downhill.”

The goal in Min LP models is to find the corner of the feasible region that lies on the lowest-valued objective function contour that still intersects

the feasible region.

Minimization

Model

Minimization

Model

In a Max model, the objective function contours are called

isoprofit

isoprofit (or profitprofit) lines.

In a Min model, the objective function contours are called

isocost

isocost (or costcost) lines.

Minimization

Model

Minimization

Model

Let’s look at a simple Min model in two decision variables (X₁ and X₂).

Min X₁ + 2X₂ s.t. -3X₁ +2X₂ < 6 X₁ + X₂ < 11 -X₁ + 3X₂ > 6 X₁ , X₂ > 0

Enter the objective function and constraints into the appropriate edit fields.

The optimal solution lies at the intersection of the

X₂ axis and constraint (3).

The optimal solution that minimizes the objective function is: 0 of X₁ 2 of X₂

Unbounded

Models

Unbounded

Models

important constraints have been left out of the model.

In this case, it results in an infinite number of allowable values for the decision variables that will improve the objective value.

To illustrate this, remove the first four

constraints from the Oak Products model and try to solve it using GLP.

After clicking AutoMax, GLP displays a message that the model is unbounded.

u n b o u n d ed a re a

Infeasible

Models

Infeasible

Models

This means that there is no combination of values for the decision variables that

simultaneously satisfies all the constraints. To illustrate this, consider the following LP model:

Max 50E + 40F s.t. E + F < 5 E - 3F < 0 10E + 15F < 150 20E + 10F < 160 30E + 10F > 135 E + F > 0

In GLP, enter the constraints and objective function. Next, click AutoMax to solve.

After clicking AutoMax, GLP displays a message that the model is infeasible.

Infeasibility depends solely on the constraints

and has nothing to do with the objective function. Note the direction (i.e., < or > that each

constraint is pointing. As you can see, there is no one region that satisfies all the constraints.

Important

Notes

Important

Notes

Every linear program will fall into one of the following three mutually exclusive categories:

1. The model has an optimal solution.

2. There is no optimal solution, because the model is unbounded.

3. There is no optimal solution, because the model is infeasible.

Sensitivity

Analysis

Sensitivity

Analysis

How sensitive is the

optimal solution to any inexact data?

The answer to this question will help

determine the credibility of the model’s recommendations.

For example, if the OV changes very little with large changes in the value of a particular

parameter, you would not be concerned about uncertainty in that value.

However, if the OV varies wildly with small

changes in that parameter, you cannot tolerate uncertainty in its value.

Sensitivity

Analysis

Sensitivity

Analysis

sensitivity analysis

sensitivity analysis (post optimality post optimality analysis

analysis) is the concept of the partial derivative, where all variables are held constant except for one. This is also known in economics as marginal analysismarginal analysis.

Sensitivity analysis is based on the proposition that all parameter values except for one number in the model are held fixed.

The graphical approach is ideal for determining how changes in two-decision-variable models affects the solution.

Sensitivity

Analysis

Sensitivity

Analysis

Changes in the Objective

Changes in the Objective

Function Coefficients

Function Coefficients

Suppose that the constraint data remain unchanged and only the

objective coefficients are changed. The only effect on the model, is that the slope of the isoprofit line will change.

By changing the coefficients of the Oak Products model’s objective function, you can see that some of those changes will not necessarily change the optimal solution, even though the isoprofit lines will have a different slope.

For example, change the objective function from

56C + 40M to 56C + 48M

Only the slope of the isoprofit line changes. However, the optimal values of

C =130 and M = 60 do not change because the same corner point solution reappears.

Alternative optimal solutions

Alternative optimal solutions

occur when the contour of the objective function is parallel to a constraint.

Alternative

Optimal

Solution

Alternative

Optimal

Solution

In the case of a two decision variable model, when the contour of the objective function is parallel to a constraint, there are two optimal corner solutions, and in fact, all the points on the constraint between the two corners are also optimal solutions.

If there is more than one solution to an LP (i.e., there exist alternative optimal solutions) then there are an infinite number of alternative

To observe the connections among the Payoff slope, corner points, and any movement of the optimal

solution from one corner point to another, right-click on the Payoff line in GLP and drag it.

For example, observe the change in the objective function coordinates and OV as the line is dragged down.

Use Solver’s Sensitivity ReportSensitivity Report feature to observe what will happen when the isoprofit line changes.

Return to the Simplified Oak Products model in the Excel spreadsheet.

After running Solver, the Solver Results dialog will open in which you can select the Sensitivity Report.

The solution gives 40 as the Objective Coefficient for Mates chairs.

Holding all other data in the model constant, the

Mate Objective Coefficient can increase by up to 16 additional dollars of unit profit and still not change the original LP corner point solution.

In general, the Allowable Increase and Allowable

Decrease entries indicate how much a given decision variable’s Objective Coefficient may change, holding all the other data in the model constant, and still

When a coefficient is changed by less than the allowable amounts, the current optimal solution remains the unique optimal solution.

For a Max model, when a coefficient is increased by its allowable amount exactly, there will be an alternative optimal corner solution with a larger optimal value for the distinguished variable.

For a MIN model, increasing a coefficient by the allowable amount exactly will produce an

alternative optimum corner with a lower optimal value for the distinguished variable.

When a variable’s coefficient is decreased by its allowable amount, there will be another

alternative optimal corner solution with the distinguished variable having a lower (MAX model) or higher (MIN model) optimal value.

A value of 0 in the Allowable Increase/Decrease

sections of the Sensitivity Report indicates that there is at least one alternative optimal corner point solution to the model.

Consider the following example:

The objective function contour is parallel to the second constraint. X3 X1 3 2 1 I

II _{Corners I and II are}

alternative optimal. Solver will find only one of these two corners as an optimal solution and the Sensitivity Report will apply only to that corner. The Sensitivity Report

would report a 0 value in

In a MaxMax model, increasing the profitability of an activity associated with a decision and keeping all other data unchanged cannot reduce the

optimal level of that activity.

For a MinMin model, increasing the cost of an activity associated with a decision and keeping all other data unchanged cannot increase the optimal level of that activity.

Sensitivity

Analysis

Sensitivity

Analysis

Ignoring the objective function, let’s observe the effects of right-hand-side changes for inequality constraints.

You can use GLP’s graphical analysis to explain the effects of changes in these parameters.

You can see the effects of changing the right-hand-side of the inequality by changing that

number directly, by dragging the constraint line or by clicking on the buttons next to each constraint in GLP.

Note that:

Tightening an inequality constraint means making it more difficult to satisfy.

For a > constraint, this means increasing the RHS. For a < constraint, this means decreasing the RHS.

Loosening an inequality constraint means making it easier to satisfy.

For a < constraint, this means increasing the RHS. For a > constraint, this means decreasing the RHS.

Tightening either contracts the feasible region or leaves it unaffected.

Loosening either expands the feasible region or leaves it unaffected.

Sensitivity

Analysis

Sensitivity

Analysis

RHS Sensitivity and the

RHS Sensitivity and the

Shadow Price

Shadow Price

For the Oak Products Model, first hold all numbers fixed except for the

inventory of Long Dowels.

Since this constraint is the < form, increasing the RHS results in “loosening” the constraint, making it easier to satisfy.

Let us change the inventory from L=1280 to L=1281, L=1320, and L=1350.

The geometric interpretation is that the feasible region is expanding.

L=1281

Note that the constraints on Legs and Long Dowels continue to be binding.

The optimal solution is C=130.25 and M=59.75

For a maximum profit of $9684 (an increase of $4, the shadow shadow price price).

The shadow priceshadow price of a given constraint is the change per unit increase in RHS with all other data held fixed.

It is called a shadow priceshadow price because its value is masked or shadowed until the model is

optimized and sensitivity analysis is done by Solver. In economic theory, the shadow price is sometimes called a reservation price.

When L = 1280, the shadow price of 4 is valid for an allowable increase (in L) of 40 Long Dowels

and an allowable decrease of 180 Long Dowels. For L values between 1000 and 1320 Long

Dowels, the change in the OV for each unit of RHS inventory increase, with all other data held fixed, is $4/Long Dowel.

L=1320

Now three constraints are binding.

The optimal solution is C=140 and M=50

For a maximum profit of $9840.

Here is the corresponding Solver Solution:

Again, note 3 binding constraints and only 2 positive decision variables.

When an LP solution has more binding constraints than positive variables, it is called degeneratedegenerate.

Here is the corresponding Sensitivity Report:

Degeneracy can lead to some anomalies when interpreting this report. Note the allowable

When L = 1320, the shadow price remains at 4, but the allowable increase is 0, which means that the value 4 does not apply to RHS values any

larger than 1320.

Indeed, the geometric analysis shows that the constraint becomes non-binding and redundant when L > 1320.

Small changes in the RHS of a non-binding

constraint cannot affect the OV, and hence for a non-binding constraint the shadow price will

The interpretation of the shadow price is valid only within a range for the given RHS. This

range is specified by the Allowable Increase

and Allowable Decrease columns in the

Constraints section of the Sensitivity Report.

L=1350

The binding constraints are now Legs and Heavy Seats.

The optimal solution is C=140 and M=50

For a maximum profit of $9840.

Here is the corresponding Solver Solution:

The Long Dowels constraint is now non-binding. The shadow price is now $0.

When L = 1350, we see that now, with the relevant constraint non-binding, the shadow price is indeed zero and the allowable increase is infinite. That is, for any further increase in L, the constraint will

remain non-binding and the shadow price will remain at the value 0.

Note that:

The shadow price of a non-binding constraint will always be zero. A non-binding constraint means that the constraint has slack or surplus.

The RHS sensitivity information that the

Sensitivity Report provides does not tell us

how the optimal decisions for C and M change. It merely explains the way in which the OV

Shortcomings of Solver’s Sensitivity Report

1. The report gives sensitivity information for perturbations of parameters only in the

immediate neighborhood of the solution, and only for changing one parameter at a time.

2. The report gives sensitivity only for effects upon the OV.

3. The report gives no sensitivity information for changes in the model’s technical

End of Part 1