Sensitivity analysis
(continued)
BSAD 30
Dave Novak
Spring 2017
Source: Anderson et al., 2013
Quantitative Methods for Business 12th
edition – some slides are directly from
J. Loucks © 2013 Cengage Learning
Overview

Sensitivity analysis continued

Changes in objective function coefficients
• Called “Range of optimality” (ROO)

Changes in RHS values
• Called “Range of feasibility” (ROF)
Focus on ROF
 Interpretation of shadow prices
 Example

Sensitivity analysis

Recall that last class we discussed:


changes in objective function coefficients
(identifying the range of optimality - ROO)
this was a detailed discussion
Discuss sensitivity analysis using the
maximization example from Lecture 7
Sensitivity analysis

Sensitivity analysis involves changing only
one coefficient at a time!


All other coefficients are held constant
We cannot use basic sensitivity analysis to
examine simultaneous changes or changes
in LHS constraint coefficients
Changes in OF coefficients

Identify the Range of Optimality (ROO)
Changes in OF coefficients do NOT impact
feasible region
 At what point do changes in the per unit
profits associated with the items we are
producing result in a change in the optimal
quantities we produce?

Sensitivity report (from
Lecture 7)
Variable Cells
Final
Xi values
ci
Final Reduced Objective Allowable Allowable
Cell
Name
Value Cost Coefficient Increase
Decrease
$B$1 X1 (# of units of product 1)
5
0
5
2 0.333333333
$B$2 X2 (# of units of product 2)
3
0
7
0.5
2
Constraints
Cell
Name
$B$10 3) Constraint #3 LHS
$B$8 1) Constraint#1 LHS
$B$9 2) Constraint #2 LHS
Final Shadow Constraint Allowable Allowable
Value Price
R.H. Side
Increase
Decrease
8
1
8 0.333333333 1.666666667
5
0
6
1E+30
1
19
2
19
5
1
Changes in OF coefficients
Each OF coefficient can take on any value
within the ROO, and the current optimal
solution will remain optimal
 ROO for c1: 4.67 ≤ c1 ≤ 7
 ROO for c2: 5 ≤ c2 ≤ 7.50
 Changes in OF coefficients will result in
changes to the final profit maximizing value
or final cost minimizing value even if the
current optimal solution remains optimal

Changes in OF coefficients
If c1 were to increase by $1.50 (from $5 /
unit to $6.50 / unit), we would still produce 5
units of x1 and 3 units of x2 because the
change to c1 is within the ROO
 Our total profit is now $53.50 instead of $46

Old c1 = $5: 5(5) + 7(3) = 46
 New c1 = $6.5: 6.5(5) + 7(3) = 53.5

Sensitivity analysis
Changes in the RHS of the constraint
coefficients impact the feasible region as
well as the optimal solution
 Identify the Range of Feasibility (ROF)
 Considering one change at a time!
 Changes to RHS values have nothing to do
with the optimal solution
 ROF gives us different information than
ROO

Original Feasible Region
x2
Coincides with Con 3: x1 + x2 < 8
8
7
Coincides with Con 1: x1 < 6
6
5
4
3
2
Con 2: 2x1 + 3x2 < 19
Feasible
Region
1
1
2
3
4
5
6
7
8
9
10
x1
Change to RHS of Con#2
x2
Con 3: x1 + x2 < 8
8
Con 1: x1 < 6
7
6
Think about a ONE unit
increase in RHS of Con
2 from 19 to 20 units
5
4
3
2
Feasible
Region
Con 2: 2x1 + 3x2 < 20
1
1
2
3
4
5
6
7
8
9
10
x1
Graphical change in RHS
x2 Con 3: x1 + x2 < 8
8
7
The values of
two critical
points change!
(x1 , x2)
(0, 6.67)
6
5
(4, 4)
4
3
2
Con 1: x1 < 6
Feasible
Region
Con 2: 2x1 + 3x2 < 20
1
1
2
3
4
5
6
7
8
9
10
x1
Changes in RHS constraint
coefficients

When examining questions related to
changes to RHS constraint coefficients you
need to determine the ROF
Inside ROF
1. Shadow price holds (is valid), AND
2. The new optimal solution (the solution that
involves the change in the RHS) will
involve the same set of binding constraints
as the old optimal solution (the original
problem before changing RHS)

Changes in RHS constraint
coefficients
Outside of the ROF
1. The interpretation of the shadow price
does not hold (doesn’t mean anything),
AND
2. The new optimal solution will involve a
different set of binding constraints

Changes in RHS constraint
coefficients
When the RHS of a constraint changes,
you typically must re-formulate and resolve the LP to obtain the new optimal
solution
 Changes to RHS constraint coefficient
values involves examining changes to the
quantity of resources that are available

The feasible region will change
 Can result in a different set of extreme
points and a different optimal solution

Sensitivity report (from
Lecture 7)
Variable Cells
Final Reduced Objective Allowable Allowable
Cell
Name
Value Cost Coefficient Increase
Decrease
$B$1 X1 (# of units of product 1)
5
0
5
2 0.333333333
$B$2 X2 (# of units of product 2)
3
0
7
0.5
2
Constraints
Cell
Name
$B$10 3) Constraint #3 LHS
$B$8 1) Constraint#1 LHS
$B$9 2) Constraint #2 LHS
Final Shadow Constraint Allowable Allowable
Value Price
R.H. Side
Increase
Decrease
8
1
8 0.333333333 1.666666667
5
0
6
1E+30
1
19
2
19
5
1
bi
Sensitivity report
The number in the “Final Value” column associated with each constraint is
the amount of the resource that is used in the optimal solution. Note that the
difference between the values in the “Final Value” column and The
“Constraint RHS” is the value of the slack or surplus variables
Variable Cells
Final Reduced Objective Allowable Allowable
Cell
Name
Value Cost Coefficient Increase
Decrease
$B$1 X1 (# of units of product 1)
5
0
5
2 0.333333333
$B$2 X2 (# of units of product 2)
3
0
7
0.5
2
Constraints
Cell
Name
$B$10 3) Constraint #3 LHS
$B$8 1) Constraint#1 LHS
$B$9 2) Constraint #2 LHS
Final Shadow Constraint Allowable Allowable
Value Price
R.H. Side
Increase
Decrease
8
1
8 0.333333333 1.666666667
5
0
6
1E+30
1
19
2
19
5
1
Shadow prices

The shadow price is a measure of the
relative value of a resource
The shadow price is typically interpreted as
the marginal value of a resource or the
maximum amount you should be willing to
pay for one additional unit of a resource
 The shadow price can be viewed as the
monetary change in the final OF value when
we increasing the resource by one unit

Shadow prices

As the RHS of a constraint increases or
decreases, other constraints may become
binding and impact the optimal solution, so
the shadow price interpretation is only
applicable for SMALL changes in the RHS –
the range of those changes is the ROF
Sensitivity report
Variable Cells
Final Reduced Objective Allowable Allowable
Cell
Name
Value Cost Coefficient Increase
Decrease
$B$1 X1 (# of units of product 1)
5
0
5
2 0.333333333
$B$2 X2 (# of units of product 2)
3
0
7
0.5
2
Constraints
Cell
Name
$B$10 3) Constraint #3 LHS
$B$8 1) Constraint#1 LHS
$B$9 2) Constraint #2 LHS
Final Shadow Constraint Allowable Allowable
Value Price
R.H. Side
Increase
Decrease
8
1
8 0.333333333 1.666666667
5
0
6
1E+30
1
19
2
19
5
1
Changes in RHS constraint
coefficients
If we increase the RHS of constraint #2 by 1
unit what are the implications with respect to
our profit and the current optimal solution?
 We now have additional resources
available with respect to constraint #2
 Is the change within the ROF?

Sensitivity report
Variable Cells
Final Reduced Objective Allowable Allowable
Cell
Name
Value Cost Coefficient Increase
Decrease
$B$1 X1 (# of units of product 1)
5
0
5
2 0.333333333
$B$2 X2 (# of units of product 2)
3
0
7
0.5
2
Constraints
Cell
Name
$B$10 3) Constraint #3 LHS
$B$8 1) Constraint#1 LHS
$B$9 2) Constraint #2 LHS
Final Shadow Constraint Allowable Allowable
Value Price
R.H. Side
Increase
Decrease
8
1
8 0.333333333 1.666666667
5
0
6
1E+30
1
19
2
19
5
1
Constraint #2: We can increase the RHS up to 5 and decrease the RHS by 1
Lower bound = 19 -1
(current RHS – allowable decrease)
Upper bound = 19 + 5 (current RHS + allowable increase)
The ROF for constraint #2 is (18 ≤ b2 ≤ 24) – within this range, the shadow price
remains valid and the optimal solution involves the same set of binding
constraints (#2 and #3)
Sensitivity report
Variable Cells
Final Reduced Objective Allowable Allowable
Cell
Name
Value Cost Coefficient Increase
Decrease
$B$1 X1 (# of units of product 1)
5
0
5
2 0.333333333
$B$2 X2 (# of units of product 2)
3
0
7
0.5
2
Constraints
Cell
Name
$B$10 3) Constraint #3 LHS
$B$8 1) Constraint#1 LHS
$B$9 2) Constraint #2 LHS
Final Shadow Constraint Allowable Allowable
Value Price
R.H. Side
Increase
Decrease
8
1
8 0.333333333 1.666666667
5
0
6
1E+30
1
19
2
19
5
1
Constraint #2: if we increase the RHS value from 19 to 20 units, our OF value
(or profit) will increase by $2 (from $46 to $48)
If we decrease the RHS value from 19 to 18 units, our OF value (or profit)
will decrease by $2 (from $46 to $44)
Changes in RHS constraint
coefficients
If we increase the RHS of constraint #2 by 1
unit what are the implications with respect to
our profit and the current optimal solution?
 Change IS within the ROF

Increasing the RHS of constraint #2 by 1 unit
(from 19 to 20) will increase our total profit
from $46 to $48 (a marginal increase of $2)
 The optimal solution will involve the same
binding constraints #2 and #3

Changes in RHS constraint
coefficients
We know that the optimal solution will
change because the feasible region has
changed and the point (5,3) is no longer
optimal (or even a critical point)
 We don’t necessarily know exactly how the
optimal solution will change (unless it’s a
graphical problem, we can’t see the extreme
points)
 We need to reformulate and resolve the
problem with the RHS of constraint #2 = 20

Graphical change in RHS
x2
Coincides with Con 3: x1 + x2 < 8
8
7
Coincides with Con 1: x1 < 6
6
5
4
3
2
Con 2: 2x1 + 3x2 < 19
Feasible
Region
1
1
2
3
4
5
6
7
8
9
10
x1
Graphical change in RHS
x2 Con 3: x1 + x2 < 8
8
7
The values of
two critical
points change!
(x1 , x2)
(0, 6.67)
6
5
(4, 4)
4
3
2
Con 1: x1 < 6
Feasible
Region
Con 2: 2x1 + 3x2 < 20
1
1
2
3
4
5
6
7
8
9
10
x1
Reformulate and resolve

Example of how this is done
Shadow prices

The Sensitivity Report provides limited
information on how changes in the RHS of
each constraint will impact the final profit
maximizing value associated with the
optimal solution as well as the feasible
region
Shadow prices

Some words of caution regarding
interpretation of shadow prices

The shadow price is generally only
applicable for small increases in the RHS
(the range of change is given by the ROF for
each constraint)
• As more resources are available and the RHS
value increases, different sets of constraints
become binding and change the optimal solution
mix (the optimal values for x1 and x2)
Shadow prices

The shadow price for a non-binding
constraint is always ZERO
The logic: if a constraint is non-binding that
means that the constraint is not part of the
current optimal solution (there is some slack
or surplus associated with the constraint)
 If the constraint is not part of the optimal
solution, then the marginal value associated
with that resource is 0

Sensitivity report
Variable Cells
Final Reduced Objective Allowable Allowable
Cell
Name
Value Cost Coefficient Increase
Decrease
$B$1 X1 (# of units of product 1)
5
0
5
2 0.333333333
$B$2 X2 (# of units of product 2)
3
0
7
0.5
2
Constraints
Cell
Name
$B$10 3) Constraint #3 LHS
$B$8 1) Constraint#1 LHS
$B$9 2) Constraint #2 LHS
Final Shadow Constraint Allowable Allowable
Value Price
R.H. Side
Increase
Decrease
8
1
8 0.333333333 1.666666667
5
0
6
1E+30
1
19
2
19
5
1
Consider constraint #1: It is non-binding, and is not part of the optimal solution
What is the marginal value of increasing the RHS value of constraint #1 by one unit?
Shadow prices

Con 1: Shadow price is 0


This makes since, as x1 < 6 is a non-binding
constraint
ROF con 1: (5 ≤ b1 ≤ ∞)

How can the allowable increase be ∞?
• In the optimal solution, the amount of resource x1
that is currently used is 5, which is less than the
amount that is currently available (x1 < 6)
• Therefore, increasing the amount of the resource
that is available will not impact the optimal
solution in any way
Graphical look at shadow
price
Current optimal solution
x2
Con 3: x1 + x2 < 8
is (5, 3) and occurs at
the intersection of Con
2 and Con 3
8
7
Con 2: 2x1 + 3x2 < 19
Con 1 has a slack value
of “1” at current optimal
solution
6
5
4
3
2
(5, 3)
What does this mean?
Feasible
Region
Con 1: x1 < 6
1
1
2
3
4
5
6
7
8
9
10
x1
Graphical look at shadow
Increasing the RHS of
price
Con 1 (even infinitely)
x2
Con 3: x1 + x2 < 8
has NO impact on the
optimal solution
8
7
Con 2: 2x1 + 3x2 < 19
6
5
4
3
2
We have 6 and only
use 5
If we had 8, we would
still use only 5 and it
would not change the
optimal solution
(5, 3)
Feasible
Region
Con 1: x1 < 8
1
1
2
3
4
5
6
7
8
9
10
x1
Shadow prices

Con 1:

The allowable decrease is 1

Further reducing the amount of the resource
available (below 5 units) will change the
optimal solution as Con 1 will become
binding at values below 5
Graphical look at shadow
price
x2
Con 1 can be decreased by
1 unit without impacting the
optimal solution
Con 3: x1 + x2 < 8
8
7
Con 2: 2x1 + 3x2 < 19
We have 5 and use 5
6
If we further decrease Con
1, our feasible region
becomes smaller and the
current optimal solution is
no longer feasible
5
4
3
2
(5, 3)
Feasible
Region
Con 1: x1 < 5
1
1
2
3
4
5
6
7
8
9
10
x1
Shadow prices
Con 2: Shadow price is 2
 The marginal value of the resource is $2
within the ROF for con 2: (18 ≤ b2 ≤ 24)

Allowable increase is 5
 Allowable decrease is 1
 This interpretation of the shadow price is
applicable only within the ROF
 Optimal solution will involve binding
constraints #2 and #3 as long as any
changes in b2 occur within the ROF (18-24)

Shadow prices
Con 3: Shadow price is 1
 The marginal value of the resource is $1
within the ROF for con 3: (6.33 ≤ b3 ≤ 8.33)

Allowable increase is 0.33
 Allowable decrease is 1.67
 This interpretation of the shadow price is
applicable only within the ROF
 Optimal solution involves binding constraints
#2 and #3 as long as any changes in b3
occur within ROF(6.33 – 8.33)

Shadow prices
Graphically, the ROF is determined by
finding the values of the RHS for each
constraint such that the same two
constraints that determined the original
optimal solution continue to determine the
new optimal solution for the problem
 ROF gives us the range of changes to the
RHS of any one of the 3 constraints, so that
the optimal solution still lies at the
intersection of constraints #2 and #3

Shadow prices

This does not mean that the optimal solution
won’t change within the ROF, just that the
same set of binding constraints (currently #2
and #3) will remain binding for changes to
the RHS as long as changes occur within
the ROF

Within the ROF, the interpretation of the
shadow price holds (shadow prices do not
mean anything outside of ROF)
Shadow prices
For changes to the RHS outside of the ROF
the problem must be reformulated and
resolved to find the new shadow prices
 Only one RHS value can be changed at a
time – does not apply to multiple
simultaneous changes
 Does not apply to changes in LHS
constraint coefficients

Sensitivity report
Variable Cells
Final Reduced Objective Allowable Allowable
Cell
Name
Value Cost Coefficient Increase
Decrease
$B$1 X1 (# of units of product 1)
5
0
5
2 0.333333333
$B$2 X2 (# of units of product 2)
3
0
7
0.5
2
Constraints
Cell
Name
$B$10 3) Constraint #3 LHS
$B$8 1) Constraint#1 LHS
$B$9 2) Constraint #2 LHS
Final Shadow Constraint Allowable Allowable
Value Price
R.H. Side
Increase
Decrease
8
1
8 0.333333333 1.666666667
5
0
6
1E+30
1
19
2
19
5
1
Assume the RHS of constraint #3 increases by 2 units.
The ROF for constraint #3 is (6.33 ≤ b3 ≤ 8.33) – within this range, the shadow price
remains valid and the same set of constraints remain binding
Change is outside of the ROF – the report tells us nothing. Reformulate and resolve
Olympic bike example
Olympic Bike is introducing two new lightweight bicycle
frames, the Deluxe (x1) and the Professional (x2)
Both frames are made from different combinations of
special aluminum and steel alloys. The anticipated unit
profits are $10 (c1) for the Deluxe and $15 (c2)
for the Professional
A supplier delivers 100 lbs. of aluminum alloy and 80 lbs.
of steel alloy per week. Each Deluxe frame requires 2lbs.
of aluminum and 3 lbs. of steel. Each Professional frame
requires 4 lbs. of aluminum and 2 lbs. of steel
How many of each type of frame should Olympic produce
each week?
Olympic bike example
Olympic bike example
Olympic bike example
X1 (# of Deluxe frames)
X2 (# of Professional frames)
15
17.5
Objective Function (Maximize Profit)
Constraints
ST:
1) Constraint#1 (materials (aluminum) constraint)
1) Constraint#1 (materials (steel) constraint)
412.5
LHS
RHS
100
80
100
80
Olympic bike example

What happens if the per unit profit for the
Deluxe frame (x1) changes from $10 (c1) to
$20 – is the current solution still optimal?
Olympic bike example
Variable Cells
Cell
Name
$B$1 X1 (# of Deluxe frames)
$B$2 X2 (# of Professional frames)
Final Reduced Objective Allowable Allowable
Value Cost Coefficient Increase Decrease
15
0
10
12.5
2.5
17.5
0
15
5 8.333333333
Constraints
Cell
Name
$B$8 1) Constraint#1 (materials (aluminum) constraint) LHS
$B$9 2) Constraint#2 (materials (steel) constraint) LHS
Final Shadow Constraint Allowable Allowable
Value Price
R.H. Side Increase Decrease
100
3.125
100
60 46.66666667
80
1.25
80
70
30
The current solution of 15 Deluxe and 17.5 Professional frames remains optimal
As long as the OF coefficient for x1 (c1) is between $7.50 and $22.50
So, increasing c1 by $10 (from $10 to $20) is within the range of optimality, so
the current optimal solution mix DOES NOT change – we will still produce 15 Deluxe
and 17.5 Professional frames
The profit maximizing solution will change from $412.50 to $562.50 because we
Are gaining $10 additional $ for each Deluxe frame we produce (= 20(15)+15(17.5))
Olympic bike example

What happens if the per unit profit for the
Deluxe frame (c1) decreases from $10 to $6
– is the current solution still optimal?
Olympic bike example
Variable Cells
Cell
Name
$B$1 X1 (# of Deluxe frames)
$B$2 X2 (# of Professional frames)
Final Reduced Objective Allowable Allowable
Value Cost Coefficient Increase Decrease
15
0
10
12.5
2.5
17.5
0
15
5 8.333333333
Constraints
Cell
Name
$B$8 1) Constraint#1 (materials (aluminum) constraint) LHS
$B$9 2) Constraint#2 (materials (steel) constraint) LHS
Final Shadow Constraint Allowable Allowable
Value Price
R.H. Side Increase Decrease
100
3.125
100
60 46.66666667
80
1.25
80
70
30
The current solution of 15 Deluxe and 17.5 Professional frames remains optimal
As long as the OF coefficient for x1 (c1) is between $7.50 and $22.50
So, decreasing c1 by $4 (from $10 to $6) is OUTSIDE the range of optimality,
and the solution mix WILL change – the current optimal solution WILL NOT
remain optimal. We have to reformulate and re-solve the problem to identify
the new optimal solution
Olympic bike example

What is the maximum amount Olympic
should pay for 50 extra lbs. of aluminum
alloy?
Olympic bike example
Variable Cells
Cell
Name
$B$1 X1 (# of Deluxe frames)
$B$2 X2 (# of Professional frames)
Final Reduced Objective Allowable Allowable
Value Cost Coefficient Increase Decrease
15
0
10
12.5
2.5
17.5
0
15
5 8.333333333
Constraints
Cell
Name
$B$8 1) Constraint#1 (materials (aluminum) constraint) LHS
$B$9 2) Constraint#2 (materials (steel) constraint) LHS
Final Shadow Constraint Allowable Allowable
Value Price
R.H. Side Increase Decrease
100
3.125
100
60 46.66666667
80
1.25
80
70
30
The shadow price is interpreted as the marginal value of an extra pound (lb.) of
aluminum (we already have paid for 100 lbs.) – what is the value of 50 additional lbs.?
Shadow price of constraint #1 (aluminum constraint) = $3.125 per lb. The ROF for
Con #1 is (53.3 ≤ b1 ≤ 160). Since the allowable increase is 60, the shadow price
interpretation holds for 50 additional lbs.
The value of 50 additional lbs. of aluminum alloy is 50 * $3.125 = $156.25
Non-intuitive shadow prices

Constraints with variables naturally on both
the left-hand (LHS) and right-hand (RHS)
sides often lead to shadow prices that have
a non-intuitive explanation
Olympic bike example

Let’s introduce an additional constraint

The number of Deluxe frames produced (x1)
must be greater than or equal to the number
of Professional frames produced (x2)
Olympic bike example
Max
10x1 + 15x2
s.t.
2x1 + 4x2 < 100
3x1 + 2x2 < 80
x 1 > x2
x1 > 0 and x2 > 0
Objective
Function
“Regular”
Constraints
Non-negativity
Constraints
Olympic answer report
Objective Cell (Max)
Cell
Name
$B$4 Objective Function (Maximize Profit)
Original Value Final Value
0
400
Variable Cells
Cell
Name
$B$1 X1 (# of Deluxe frames)
$B$2 X2 (# of Professional frames)
Original Value Final Value
Integer
0
16 Contin
0
16 Contin
Constraints
Cell
Name
$B$8 1) Constraint#1 (materials (aluminum) constraint) LHS
$B$9 2) Constraint#2 (materials (steel) constraint) LHS
$B$10 3) Constraint #3 (production constraint) LHS
Cell Value
Formula
Status
Slack
96 $B$8<=$C$8 Not Binding
4
80 $B$9<=$C$9 Binding
0
16 $B$10>=$C$10 Binding
0
Olympic sensitivity report
Variable Cells
Cell
Name
$B$1 X1 (# of Deluxe frames)
$B$2 X2 (# of Professional frames)
Final Reduced Objective Allowable Allowable
Value Cost Coefficient Increase
Decrease
16
0
10
12.5
25
16
0
15
1E+30 8.333333333
Constraints
Cell
Name
$B$8 1) Constraint#1 (materials (aluminum) constraint) LHS
$B$9 2) Constraint#2 (materials (steel) constraint) LHS
$B$10 3) Constraint #3 (production constraint) LHS
Final Shadow Constraint Allowable Allowable
Value Price
R.H. Side
Increase
Decrease
96
0
100
1E+30
4
80
5
80 3.333333333
80
16
-5
0 26.66666667
2.5
Olympic bike example

Interpret the shadow prices of:

Constraint #1
Olympic bike example

Interpret the shadow prices of:

Constraint #2
Olympic bike example

Interpret the shadow prices of:

Constraint #3
Olympic bike example

Interpret the range of optimality for OF
coefficient:
 c1
Olympic bike example

Interpret the range of optimality for OF
coefficient:
 c2
Reduced costs
The reduced cost of a variable is typically
the shadow price of the corresponding nonnegativity constraints
 If a decision variable has a positive value at
the optimal solution, the reduced cost for
that decision variable = 0

Reduced costs

If a decision variable has a value of 0 at the
optimal solution (produce zero of that thing),
the reduced cost for that variable ≠ 0, and
can be interpreted as:
the amount the objective value will change if
we increase the value of this variable to one
 the amount by which the objective coefficient
would have to decrease in order to have a
positive value for that variable in an optimal
solution

Summary

Introduction to sensitivity analysis
Changes in objective function coefficients
 Identify ROO

• What does this mean graphically?
• How are these changes interpreted?
Changes in RHS values
 Identify ROF

• What does this mean graphically?
• How are these changes interpreted?
Summary

Interpretation of shadow prices


Where to find these on the Sensitivity Report
Olympic bike example