What-If Analysis for Linear
Programming
Chapter 5: Hillier and Hillier
Agenda
Define What is What-If Analysis
 The Importance of What-If Analysis
 Discuss the Effect of Changing One
Coefficient in the Objective Function
 Discuss the Effect of Changing Two
Coefficients in the Objective Function
 Discuss the Effect of Single Changes in a
Constraint

What-If Analysis

Its an analysis that examines what happens
to your optimal decision when the
assumptions of your model change or are
different.
– In practice, what-if analysis consists of
changing a particular set of parameters in the
objective function or the constraints to see what
happens to the optimal solution.
Importance of What-If Analysis


In application, many aspects of the model are
based on estimations which cannot be determined
precisely.
What-if analysis is used to examine the model to
understand how sensitive it is to the parameters in
the model.
– By knowing how sensitive the model is to the
parameters, you will know which parameters you
should spend the most time on trying to estimate
correctly.
Importance of What-If Analysis

What-if analysis can be broken-up into two major
types.
– Sensitivity analysis is when you examine the changes in
the parameters of the model to see what happens to the
optimal solution.
– The second type of analysis examines when you look at
different assumptions that affect more than just the
parameters.
• This is usually done by changing the objective function and
constraints in fundamental ways beyond changing the
parameters.
Effect of Changing One Coefficient
in the Objective Function
By changing a parameter in the objective
function, you are affecting the slope of the
objective function which has the possibility
of changing your optimal solution.
 What-if analysis examines how much of a
parameter shift can be sustained before
changing the optimal solution.

Effect of Changing One Coefficient
in the Objective Function Cont.

There are two ways to examine how the
change in the parameter will affect the
optimal solution.
– The first way is to solve the problem with the
new parameter multiple times.
– The second method is to use Solver’s
Sensitivity Report to understand what
parameter changes would affect the optimal
solution.
Solving the Excel Model Multiple
Times with Multiple Parameters


Whenever you change a parameter in the model
you must tell Excel to resolve the problem by
going to Solver.
When doing this type of sensitivity analysis, you
want to change the parameters in a way that will
allow you to find the key points quickly.
– You could use some form of divide and conquer to find
the key changing points.
– You could establish a particular interval to help find the
sensitive points.
Solver Table

Solver Table is a tool developed by the
textbook authors to solve the model
multiple times using different parameters.
– The current version on your disk may not be
operable.
– How would you go about finding an operable
version?
Solver Table Cont.
Solver Table can change up to two
parameters at a time.
 In class activity: Build a sensitivity chart for
changing the prices of windows.

– Examine prices that range from $100 to $1000.
– Use the Solver Table to find the price of
windows that changes the optimal solution
from 2,6 to 4,3.
Solver’s Sensitivity Report
Solver has another way of finding the
parameters that will change the optimal
solution.
 This is done by using Solver’s Sensitivity
Report.
 To get the Sensitivity Report, you need to
highlight the report after you have used
Solver.

Solver’s Sensitivity Report Cont.
Microsoft Excel 9.0 Sensitivity Report
Worksheet: [Wyndor.xls]Wyndor
Report Created: 6/19/2002 9:53:42 AM
Adjustable Cells
Final Reduced Objective
Cell
Name
Value
Cost
Coefficient
$C$12 Units Produced Doors
2
0
300
$D$12 Units Produced Windows6
0
500
Allowable
Increase
450
1E+30
Allowable
Decrease
300
300
Allowable
Increase
1E+30
6
6
Allowable
Decrease
2
6
6
Constraints
Cell
Name
$E$7 Plant 1 Used
$E$8 Plant 2 Used
$E$9 Plant 3 Used
Final Shadow
Value
Price
2
0
12
150
18
100
Constraint
R.H. Side
4
12
18
Analyzing the Sensitivity Report

To find the range of the variable before the
optimal solution will change, you can use
the Solver information in the following way.
– The bottom end of the range on the coefficient
is:
• Objective coefficient – Allowable Decrease
– The upper end of the range of the coefficient is:
• Objective coefficient + Allowable Increase
Analyzing the Sensitivity Report
Cont.
In the Wyndor example the price of the
doors could increase to $750 or decrease to
$0 before the optimal solution would
change.
 In the Wyndor example the price of the
windows could increase an infinite amount
or decrease to $200 before the optimal
solution would change.

Sensitive Parameters

A parameter is considered a sensitive
parameter if small changes lead to a change
in the optimal solution.
– These parameters are the ones you will focus on
to make sure you have them as close to correct
as possible.
Effect of Simultaneous Changes in
the Objective Function Coefficients



In many cases, more than one parameter is
uncertain.
In this case you would like to know what would
happen to your optimal solution when multiple
parameters are different than what you expected.
Typically, this analysis can be done by changing
multiple parameters at once and seeing what
happens to the optimal solution.
Excel Side Note

You can represent a solution set in a single
cell by placing an & in front of the variable
you want to add.
– For example:="("&C12&", "&D12&")” gives
(2, 6) in the same cell.
The 100 Percent Rule for Simultaneous
changes in Objective Function Coefficients


This is a rule that tells you how much of each
constraint is allowed to change simultaneously
before the optimal might change.
This rule says that if the sum of the proportions of
parameter change divided by allowable changes in
absolute value terms of all the coefficients does
not exceed 100%, then the original optimal
solution was still be optimal.
– If it changes by more than 100%, you cannot be sure.
Calculating a Percentage
Change

The percentage change for a value from the
100% rule can be calculated as:

(New Value – Old Value) / Allowable Change

For example: when 300 changes to 600 and
the allowable change is 900 you get a
proportional change of (600-300)/900 which
equals approximately 33.33%.
The Effect of Single Changes in a
Constraint
This type of what-if analysis examines what
happens to the optimal decision when a
constraint coefficient changes.
 To examine this issue, you can methodically
change the parameter on the coefficient or
you could use the Sensitivity Report from
Solver.

Solver’s Sensitivity Report
Microsoft Excel 9.0 Sensitivity Report
Worksheet: [Wyndor.xls]Wyndor
Report Created: 6/19/2002 9:53:42 AM
Adjustable Cells
Final Reduced Objective
Cell
Name
Value
Cost
Coefficient
$C$12 Units Produced Doors
2
0
300
$D$12 Units Produced Windows6
0
500
Allowable
Increase
450
1E+30
Allowable
Decrease
300
300
Allowable
Increase
1E+30
6
6
Allowable
Decrease
2
6
6
Constraints
Cell
Name
$E$7 Plant 1 Used
$E$8 Plant 2 Used
$E$9 Plant 3 Used
Final Shadow
Value
Price
2
0
12
150
18
100
Constraint
R.H. Side
4
12
18
Shadow Price

The shadow price for a constraint is the rate
at which the value of the objective function
can be increased by increasing the righthand side of the constraint by a small
amount.
– This amount tells you what effect a change in
the constraint will have on the objective
function.
Allowable Range


The allowable range of a functional constraint is
the range of values for this right-hand side over
which this constraint’s shadow price remains
valid.
The bottom end of the range is calculated by:
– Constraint RH Side – Allowable Decrease

The upper end of the range is calculated by:
– Constraint RH Side + Allowable Increase