LINEAR PROGRAMMING
SENSITIVITY ANALYSIS
Learning Objectives

Learn sensitivity concepts

Understand, using graphs, impact of changes in objective function
coefficients, right-hand-side values, and constraint coefficients on
optimal solution of a linear programming problem.

Generate answer and sensitivity reports using Excel's Solver.

Interpret all parameters of reports for maximization and
minimization problems.

Analyze impact of simultaneous changes in input data values
using 100% rule.

Analyze the impact of addition of new variable using pricing-out
strategy.
Introduction (1 of 2)
 Optimal solutions to LP problems have been examined under
deterministic assumptions.
 Conditions in most real world situations are dynamic and
changing.
 After an optimal solution to a problem is found, input data
values are varied to assess optimal solution sensitivity.
 This process is also referred to as sensitivity analysis or postoptimality analysis.
Introduction (2 of 2)
 Sensitivity analysis determines the effect on optimal solutions of
changes in parameter values of the objective function and
constraint equations
 Changes may be reactions to anticipated uncertainties in the
parameters or the new or changed information concerning the
model
The Role of Sensitivity Analysis of the
Optimal Solution
 Is the optimal solution sensitive to changes in input
parameters?
 Possible reasons for asking this question:
Parameter values used were only best estimates.
Dynamic environment may cause changes.
“What-if” analysis may provide economical and
operational information.
1. Sensitivity Analysis of
Objective Function Coefficients.
Ranges of Optimality
The value of the objective function will change if the
coefficient multiplies a variable whose value is nonzero.
The optimal solution will remain unchanged as long as:
an objective function coefficient lies within its range of optimality
there are no changes in any other input parameters.
The optimality range for an objective coefficient is the range
of values over which the current optimal solution point will
remain optimal
For two variable LP problems the optimality ranges of objective
function coefficients can be found by setting the slope of the
objective function equal to the slopes of each of the binding
constraints
Sensitivity Analysis Using Graphs
Example 1: Galaxy Industries (1 of 5)
Max 8X1 + 5X2 (Weekly profit)
subject to
2X1 + 1X2 < = 1200 (Plastic)
3X1 + 4X2 < = 2400 (Production Time)
X1 + X2 < = 800
(Total production)
X1 - X2 < = 450
(Mix)
Xj> = 0, j = 1,2
(Nonnegativity)
Sensitivity Analysis Using Graphs
Example 1:(2 of 5)
We now demonstrate
the search for an optimal solution
X2
1200
Start at some arbitrary profit, say profit = $2,000...
Then increase the profit, if possible...
...and continue until it becomes infeasible
800
Profit
4,
Profit
= $=$5040
2,
3,
000
600
X1
400
600
800
Sensitivity Analysis Using Graphs
Example 1 (3 of 5)
MODEL SOLUTION
Space Rays = 480 dozens
Zappers
= 240 dozens
Profit
= $5040
– This solution utilizes all the plastic and all the production hours.
– Total production is only 720 (not 800).
– Space Rays production exceeds Zapper by only 240 dozens (not
450).
Sensitivity Analysis Using Graphs
Example 1 (4 of 5)
1200
800
X2
The effects of changes in an objective function
coefficient
on the optimal solution
600
X1
400
600
800
Sensitivity Analysis Using Graphs
Example 1 (5 of 5)
1200
X2
Range of optimality
800
600
400
600
800
The effects of changes in an objective function coefficients
on the optimal solution
X1
Sensitivity Analysis Using Graphs
Example 2: Beaver Creek Pottery (1 of 4)
Maximize Z = $40x1 + $50x2
subject to:
1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0
Sensitivity Analysis Using Graphs
Example 2 (2 of 4)
Maximize Z = $100x1 + $50x2
subject to:
1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0
Sensitivity Analysis Using Graphs
Example 2 (3 of 4)
Maximize Z = $40x1 + $100x2
subject to:
1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0
Sensitivity Analysis Using Graphs
(Objective Function Coefficient Sensitivity Range for c1 and c2 )
Example 2 (4 of 4)
objective function Z = $40x1 + $50x2
sensitivity range for:
x1: 25  c1  66.67
x2: 30  c2  80
Objective Function Coefficient Ranges
Excel “Solver” Results Screen (1 of 3)
Objective Function Coefficient Ranges
Beaver Creek Example Sensitivity Report (2 of 3)
Objective Function Coefficient Ranges
QM for Windows Sensitivity Range Screen
Beaver Creek Example (3 of 3)
Sensitivity Analysis Using Graphs
Example 3: High Note Sound Company(HNSC) (1 of 4)
 HNSC Manufactures quality CD players and stereo receivers.
 Each product requires skilled craftsmanship.
 LP problem formulation:
Objective: maximize profit = $50C + $120R
subject to
2C + 4R  80
(Hours of electricians' time available)
3C + R  60
(Hours of audio technicians' time available)
C, R  0
(Non-negativity constraints)
Where:
C = number of CD players to make.
R = number of receivers to make.
Sensitivity Analysis Using Graphs
Example 3(2 of 4)
Sensitivity Analysis Using Graphs
Example 3 (3 of 4)
Impact of price change of Receivers
If unit profit per stereo receiver (R) increased
from $120 to $150, is corner point a still the
optimal solution? YES !
But Profit is $3,000 = 0 ($50) + 20 ($150)
Sensitivity Analysis Using Graphs
Example 3 (4 of 4)
Impact of price change of Receivers
If receiver’s profit coefficient changed from
$120 to $80, slope of isoprofit line changes
causing corner point (b) to become optimal.
But Profit is $1,760 = 16 ($50) + 12 ($80).
Objective Function Coefficient
Sensitivity Range
(for a Cost Minimization Model)
Minimize Z = $6x1 + $3x2
subject to:
2x1 + 4x2  16
4x1 + 3x2  24
x1, x2  0
sensitivity ranges:
4  c1  
0  c2  4.5
Multiple changes
– The range of optimality is valid only when a single
objective function coefficients changes.
– When more than one variable changes we turn to
the 100% rule.
The 100% Rule
1. For each increase (decrease) in an objective function
coefficient, calculate (and express as a percentage) the
ratio of the change in the coefficient to the maximum
possible increase (decrease) as determined by the
limits of the range of optimality.
2. Sum all these percent changes. If the total is less than
100 percent, the optimal solution will not change. If
this total is greater than or equal to 100%, the optimal
solution may change.
Reduced Costs
The reduced cost for a variable at its lower bound
(usually zero) yields:
The amount the profit coefficient must change
before the variable can take on a value above its
lower bound.
The amount the optimal profit will change per
unit increase in the variable from its lower bound.
2. Sensitivity Analysis of Right Hand
Side Values
Changes in Right-Hand-Side Values of
Constraints
The sensitivity range for a RHS value is the range
of values over which the quantity (RHS) values
can change without changing the solution variable
mix, including slack variables.
Sensitivity Analysis of
Right-Hand SideValues
Any change in the right hand side of a binding
constraint will change the optimal solution.
Any change in the right-hand side of a nonbinding
constraint that is less than its slack or surplus, will
cause no change in the optimal solution.
Changes in Constraint Quantity (RHS) Values
Increasing the Labor Constraint (1 of 3)
Example 1: Beaver Creek
Maximize Z = $40x1 + $50x2
subject to: 1x1 + 2x2  40
4x2 + 3x2  120
x1, x2  0
Changes in Constraint Quantity (RHS) Values
Sensitivity Range for Labor Constraint (2 of 3)
Example 1: Beaver Creek
Sensitivity range for:
30  q1  80 hr
Changes in Constraint Quantity (RHS) Values
Sensitivity Range for Clay Constraint (3 of 3)
Example 1: Beaver Creek
Sensitivity range for:
60  q2  160 lb
Constraint Quantity (RHS) Value Ranges by Computer
Excel Sensitivity Range for Constraints (1 of 2)
Example 1: Beaver Creek
Constraint Quantity (RHS) Value Ranges by Computer
QM for Windows Sensitivity Range (2 of 2)
Example 1: Beaver Creek
Sensitivity Analysis of
Right-Hand SideValues & Shadow Prices
In sensitivity analysis of right-hand sides of constraints we are
interested in the following questions:
 Keeping all other factors the same, how much would the optimal
value of the objective function (for example, the profit) change if
the right-hand side of a constraint is changed by one unit?
 For how many additional or fewer units this per unit change will
be valid?
Changes in Right-hand-side (RHS) &
Shadow Prices
RHS of Binding Constraint • If RHS of non-redundant constraint changes, size of feasible region
changes.
– If size of region increases, optimal objective function improves.
– If size of region decreases, optimal objective function worsens.
• Relationship expressed as Shadow Price.
• Shadow Price is change in optimal objective function value for one
unit increase in RHS.
Shadow Prices (Dual Values)
 Defined as the marginal value of one additional unit of
resource.
 The sensitivity range for a constraint quantity value is also
the range over which the shadow price is valid.
Range of Feasibility
 The set of right - hand side values for which same set of
constraints determines the optimal point.
 Within the range of feasibility, shadow prices remain constant;
however, the optimal solution will change.
Excel Sensitivity Report for Shadow Prices (1 of 2)
Example 1:Beaver Creek Pottery
Maximize Z = $40x1 + $50x2 subject
to:
x1 + 2x2  40 hr of labor
4x1 + 3x2  120 lb of clay
x1, x2  0
Solution Screen (2 of 2)
Example 1: Beaver Creek Pottery
Example 2: High Note Sound Company Problem
(1 of 12)
 HNSC Manufactures quality CD players and stereo receivers.
 Each product requires skilled craftsmanship.
 LP problem formulation:
Objective: maximize profit = $50C + $120R
subject to
2C + 4R  80
(Hours of electricians' time available)
3C + R  60
(Hours of audio technicians' time available)
C, R  0
(Non-negativity constraints)
Where:
C = number of CD players to make.
R = number of receivers to make.
Sensitivity Analysis of Right-hand-side (RHS) Values
Example 2: High Note Sound Company (2 of 12)
May change the
feasible region size.
May change or move
corner points.
Increase in Electricians’ Available Time
Example2: High Note Sound Company (3 of 12)
• As available electricians’ time increases,
corner points a and b will move closer to one
other.
• Further increases in available electricians’ time
may make this constraint redundant.
Decrease in Electricians’ Available Time
Example 2: High Note Sound Company (4 of 12)
• As available electricians’ time decreases, corner
points b and c move closer to one another from their
current locations.
• Corner points b and c will no longer be feasible, and
intersection of electricians’ time constraint with
horizontal (C) axis will become a new feasible corner
point.
Sensitivity Analysis Using Solver
Example 2: High Note Sound Company (5 of 12)
Solver Report
Example 2: High Note Sound Company (6 of 12)
Answer Report
Example 2: High Note Sound Company (7 of 12)
Sensitivity Report
• Sensitivity report has two distinct components.
(1) Table titled Adjustable Cells
(2) Table titled Constraints.
• Tables permit one to answer several "what-if" questions
regarding problem solution.
• Consider a change to only a single input data value.
• Sensitivity information does not always apply to
simultaneous changes in several input data values.
Sensitivity Report
Example 2: High Note Sound Company (8 of 12)
 Primary information is provided by Shadow Price
 Resources available:
 80 hours of electricians’ time.
 60 hours of audio technicians’ time.
 Final Values in table reveal optimal solution requires:
 all 80 hours of electricians’ time.
 Only 20 hours of audio technicians’ time.
 Electricians’ time constraint is binding.
 Audio technicians’ time constraint is non-binding.
 40 unused hours of audio technicians’ time are referred to as
slack.
Changes in Right-Hand-Side (RHS)
Example 2: High Note Sound Company (9 of 12)

In case of electrician hours Shadow Price is $30.

For each additional hour of electrician time that

firm can increase profits by $30.
Changes in RHS of a Non-binding Constraint
Example 2: High Note Sound Company (10 of 12)
 Audio technicians’ time has 40 unused hours.
 No interest in acquiring additional hours of resource.
 Shadow price for audio technicians’ time is zero.
 Allowable increase for RHS value is infinity (shown as 1E+30
by Solver).
 Once 40 hours is lost (current unused portion, or slack) of
audio technicians’ time, resource also becomes binding.
 Any additional loss of time will clearly have adverse effect on
profit.
Changes in the Objective Function Coefficient (OFC)
Example 2: High Note Sound Company (11 of 12)
Adjustable Cells
 Reduced Cost value - shows amount one will ‘lose’ if

solution is forced to make an additional unit.
 Current value is 0. If one makes 1, firm will lose $10.
 Allowable Increase - indicates if price increases by $10,
one will profit by making additional CDs.
 Allowable Decrease – infinity (1E+30) indicates if $50
is not attractive enough to make CD – any price below
it will not make it attractive either!
Changes in Objective Function Coefficient (OFC)
Example 2: High Note Sound Company Example (12 of 12)
Sensitivity Analysis For A Larger
Maximization Example
Example 3: Anderson Electronics (1 of 13)
Considering producing four potential products: VCRs, stereos,
televisions (TVs), and DVD players:
Profit per unit:
VCR
Stereo
TV
DVD
$41
$32
$72
$54
LP Formulation
Example 3: Anderson Electronics (2 of 13)
Objective: maximize profit =
$29 V + $32 S + $72 T + $54 D
subject to
3 V + 4 S + 4 T + 3 D  4700
2 V + 2 S + 4 T + 3 D  4500
1 V + 1 S + 3 T + 2 D  2500
V, S, T, D  0
Where: V
S
T
D
=
=
=
=
(Electronic components)
(Non-electronic components)
(Assembly time in hours)
number of VCRs to produce.
number of Stereos to produce.
number of TVs to produce.
number of DVD players to produce.
Excel Solver Set-up and Solution
Example 3: Anderson Electronics Example (3 of 13)
Excel Solver Answer Report
Example 3: Anderson Electronics (4 of 13)
Excel Solver Sensitivity Report
Example 3: Anderson Electronics (5 of 13)
Excel Solver Sensitivity Report
Example 3: Anderson Electronics (6 of 13)
Adjustable Cells
Non Zero value decision variables, Stereos and DVDs:
Produce 380 Stereos with unit profit of $32.
• Decision should not change as profit is between $31.33 and $72:
Objective Coefficient – Allocable Decrease ($32 - $1.67)
and
Objective Coefficient – Allocable Increase ($32+$40)
Produce 1060 DVDs with unit profit of $54.
• Decision should not change as profit is between $49 and $64:
Objective Coefficient – Allocable Decrease ($54 - $5)
and
Objective Coefficient – Allocable Increase
($54+$10)
Excel Solver Sensitivity Report
Example 3: Anderson Electronics (7 of 13)
Zero value decision variables, VCRs and TVs:
Produce 0 VCRs with unit cost of $1.00 (Reduced Cost).
• Decision to make 0 should not change as profit is below $29 – but
should change over and $30:
Objective Coefficient – Allocable Decrease ($29 - infinity) and
Objective Coefficient – Allocable Increase ($29 + $1).
Produce 0 TVs with unit cost of $8.00 (Reduced Cost).
• Decision to make 0 should not change as profit is below $72 – but
should change over and $80:
Objective Coefficient – Allocable Decrease ($72 - infinity) and
Objective Coefficient – Allocable Increase ($72 + $8).
Constraints on the Sensitivity Report
Example 3: Anderson Electronics (8 of 13)
Constraints on the Sensitivity Report
Example 3: Anderson Electronics (9 of 13)
Simultaneous Changes In Parameter Values
Example 3: Anderson Electronics (10 of 13)
Possible to analyze impact of simultaneous changes on optimal
solution only under specific condition:
 (Change / Allowable change)  1
• If decrease RHS from 4,700 to 4,200, allowable decrease is
950.
The ratio is: 500 / 950 = 0.5263
• If increase 200 hours (from 2,500 to 2,700) in assembly time,
allowable increase is 466.67.
The ratio is: 200 / 466.67 = 0.4285
• The sum of these ratios is:
Sum of ratios = 0.5263 + 0.4285 = 0.9548 < 1
Since sum does not exceed 1, information provided in
sensitivity report is valid to analyze impact of changes.
Simultaneous Changes In Parameter Values
Example 3:Anderson Electronics (11 of 13)
• Decrease of 500 units in electronic component
availability reduces size of feasible region and causes
profit to decrease.
– Magnitude of decrease is $1,000 (500 units x $2 per unit).
• Increase of 200 hours of assembly time results in larger
feasible region and net increase in profit.
– Magnitude of increase is $4,800 (200 hours x $24 per
hour).
• Net impact of both changes simultaneously is an
increase in profit by $3,800 ( $4,800 - $1,000).
Simultaneous Changes In Parameter Values
Example 3: Anderson Electronics (12 of 13)
Simultaneous Changes in OFC Values
• What is impact if selling price of DVDs drops by $3 per unit
and at same time selling price of stereos increases by $8 per
unit?
• For current solution to remain optimal, allowable decrease in
DVD players is $5, while allowable increase in OFC for
stereos is $40.
– Sum of ratios is:
Sum of ratios = $3 / $5 + $8 / $40 = 0.80 < 1
– $3 decrease in profit per DVD player causes total profit to decrease by
$3,180 (i.e., $3 x 1,060).
– $8 increase in unit profit of each stereo results in total profit of $3,040
(i.e., $8 x 380).
• Net impact is a decrease in profit of only $140 to a new value
of $69,260.
Checking Validity of the 100% Rule
Example 3: Anderson Electronics Example (13 of 13)
• Calculate ratio of reduction in each resource’s availability
to allowable decrease for that resource.
Sum of ratios = 5/950 + 4/560 + 4/1325 = 0.015 < 1
• Required Profit on Each HTS:
5 x shadow price of electronic components +
4 x shadow price of non-electronic components +
4 x shadow price of assembly time
or 5 x $2 + 4 x $0 + 4 x $24 = $106
• Profit contribution of each HTS has to at least make up
shortfall in profit.
• OFC for HTS must be at least $106 in order for optimal
solution to have non-zero value.
Sensitivity Analysis - Minimization Example
Example 4: Burn-Off Diet Drink Example (1 of 5)
• Plans to introduce miracle drink that will magically burn
fat away.
LP Formulation
Example 4: Burn-Off Diet Drink Example (2 of 5)
Objective: minimize daily dose cost in cents.
4A + 7B + 6C + 3D
Subject to
A + B + C + D  36
(Daily dose requirement)
3A + 4B + 8C + 10D  280
(Chemical X requirement)
5A + 3B + 6C + 6D  200
(Chemical Y requirement)
10A + 25B + 20C + 40D  1050 (Chemical Z max limit)
A, B, C, D  0
Excel Solution
Example 4: Burn-Off Diet Drink (3 of 5)
Solver Answer Report
Example 4: Burn-Off Diet Drink (4 of 5)
Solver Sensitivity Report
Example 4: Burn-Off Diet Drink (5 of 5)
Problem Statement (1 of 3)
Example Problem 5
• Two airplane parts: no.1 and no. 2.
• Three manufacturing stages: stamping, drilling, milling.
• Decision variables: x1 (number of part no.1 to produce)
x2 (number of part no.2 to produce)
• Model: Maximize Z = $650x1 + 910x2
subject to:
4x1 + 7.5x2  105 (stamping,hr)
6.2x1 + 4.9x2  90 (drilling, hr)
9.1x1 + 4.1x2  110 (finishing, hr)
x1, x2  0
Graphical Solution (2 of 3)
Example Problem 5
Maximize Z = $650x1 + $910x2
subject to:
4x1 + 7.5x2  105
6.2x1 + 4.9x2  90
9.1x1 + 4.1x2  110
x1, x2  0
s1 = 0, s2 = 0, s3 = 11.35 hr
485.33  c1  1,151.43
137.76  q1  89.10
Graphical Solution
Excel Solution (3 of 3)
Example Problem 5
Summary
• Sensitivity analysis used by management to answer series of “
what-if ” questions about LP model inputs.
• Tests sensitivity of optimal solution to changes:
– Profit or cost coefficients, and
– Constraint RHS values.
• Explored sensitivity analysis graphically (with two decision
variables).
• Discussed interpretation of information:
– In answer and sensitivity reports generated by Solver.
– In reports used to analyze simultaneous changes in model
parameter values.
– Determine potential impact of new variable in model.