Managerial Decision Modeling with
Spreadsheets
Chapter 4
Linear Programming Sensitivity Analysis
Learning Objectives
•
•
•
•
•
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 impact of addition of new variable using
pricing-out strategy.
2
4.1 Introduction
• 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 problem is
found, input data values are varied to assess
optimal solution sensitivity.
• This process is also referred to as sensitivity
analysis or post-optimality analysis.
3
4.2 Sensitivity Analysis Using Graphs
High Note Sound Company
• 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.
4
High Note Sound Company Problem Solution
5
Changes in Objective Function Coefficient
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)
6
Changes in Objective Function Coefficient
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).
7
4.3 Sensitivity Analysis Using Solver
• In Answer Report
• Final Values: objective function, decision
variables.
• Binding and nonbinding constraints
• Slack: Unused resource.
• In Sensitivity Report
• Adjustable Cells:
– Objective Function Coefficients
– Reduced Cost
– Allowable Changes
• Constraints:
– Shadow Price
– Allowable Changes
8
High Note Sound Company Answer Report
9
High Note Sound Company Answer Report
• 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.
• Binding and Non-binding Constraints:
– Electricians’ time constraint is binding.
– Audio technicians’ time constraint is non-binding.
• 40 unused hours of audio technicians’ time are referred to
as slack.
10
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.
11
High Note Sound Company Sensitivity Report
12
Changes in Constraint Right-hand-side (RHS)
• Primary information is provided by Shadow Price
• Shadow Price is change in optimal objective function
value for one unit increase in RHS.
• The shadow price is positive for binding constraints
and is zero for nonbinding constraints.
13
Changes in Right-hand-side (RHS)
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.
14
Changes in Right-hand-side (RHS)
High Note Sound Company
Constraints
Cell
Name
Final
Shadow
Constraint
Allowable
Allowable
Value
Price
R.H. Side
Increase
Decrease
$D$8
Electricians' Time
80.00
30.00
80.00
160.00
80.00
$D$9
Audio Technicians' Time
20.00
0.00
60.00
1E+30
40.00
•
In case of electrician hours, shadow price is $30.
•
For each additional hour of electrician time that
firm can increase profits by $30.
•
The range of RHS for electrician time with a
shadow price of $30 is (0, 240).
•
How to calculate shadow price and range? Excel. 15
Change in RHS of Nonbinding Constraint
Constraints
Cell
•
Name
Final
Shadow
Constraint
Allowable
Allowable
Value
Price
R.H. Side
Increase
Decrease
$D$8
Electricians' Time
80.00
30.00
80.00
160.00
80.00
$D$9
Audio Technicians' Time
20.00
0.00
60.00
1E+30
40.00
In case of audio technicians’ time, shadow price is zero.
– Audio technicians’ time has 40 unused hours.
– No interest in acquiring additional hours of resource.
– Allowable increase for RHS value is infinity.
• Allowable decrease for RHS value is 40.
– 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.
• The range of RHS for audio technicians’ time with a shadow
price of $0 is (20, infinite).
16
Change in Objective Function Coefficient (OFC)
Adjustable Cells
• Reduced Cost value - shows the difference between the
marginal contribution of a decision variable and the
marginal worth of the resources it uses.
– Objective Function Coefficients
– Shadow Prices and Resources Used
• Allowable Increase and Allowable Decrease – the
limits to which the objective function coefficient of a
decision variable can be changed without affecting the
optimality of the current solution.
17
Change in Objective Function Coefficient (OFC)
High Note Sound Company
18
Change in Objective Function Coefficient (OFC)
Reduced Cost for each CD player
• The marginal contribution is the objective coefficient
$50.
• The marginal worth of the resources used:
– Resources Used: 2 hours of electrician time and 3 hours of
audio technician’s time.
– Shadow Prices: $30 for per hour of electrician time and $0 for
per hour of audio technician time.
– Marginal Cost: 2 x $30 + 3 x $0 = $60.
• Reduced Cost: $60 - $50 = $10
• Current value is 0. If one makes 1, firm will lose $10.
19
Change in Objective Function Coefficient (OFC)
Allowable Increase and Decrease for the coefficient
of CD players
• Allowable Increase - indicates if the price of CD
players 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!
20
Change in Objective Function Coefficient (OFC)
Reduced Cost for each Stereo Receiver
• The marginal contribution is the objective coefficient
$120.
• The marginal worth of the resources used:
– Resources Used: 4 hours of electrician time and 1 hours of
audio technician’s time.
– Shadow Prices: $30 for per hour of electrician time and $0 for
per hour of audio technician time.
– Marginal Cost: 4 x $30 + 0 x $0 = $120.
• Reduced Cost: $120 - $120 = $0.
21
Change in Objective Function Coefficient (OFC)
Allowable Increase and Decrease for each Stereo
Receiver
• Allowable Increase - infinity (1E+30) indicates if $120
is profitable enough to make receiver – any price above
it will also be profitable.
• Allowable Decrease – $20 indicates if the price of
receivers drops below than $100, it is not optimal to
produce 20 receivers and no CDs.
22
4.4 Sensitivity Analysis For A Larger
Maximization Example
• Anderson Electronics
Considering producing four potential products: VCRs,
stereos, televisions (TVs), and DVD players:
Profit per unit:
VCR
Stereo
TV
DVD
$29
$32
$72
$54
Electronic Components
Non-electronic Components
Assembly time (hours)
Selling price (per unit)
VCR
Stereo
TV
DVD
Supply
Cost
3
2
4
2
4
4
3
3
4,700
4,500
$7
$5
1
$70
1
$80
2,500
$10
3
2
$150 $110
23
Anderson Electronics LP Formulation
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.
24
Excel Solver Answer Report
Target Cell (Max)
Cell
$F$8
Name
Profit
Original Value
$0.00
Final Value
$69,400.00
Adjustable Cells
Cell
Name
Original Value
Final Value
$B$5
Solution value VCR
0.00
0.00
$C$5
Solution value Stereo
0.00
380.00
$D$5
Solution value TV
0.00
0.00
$E$5
Solution value DVD
0.00
1060.00
Constraints
Cell
Name
Cell Value
Formula
Status
$F$10
Electronic comp
4700.00 $F$10<=$H$10
Binding
$F$11
Non-electronic comp
3940.00 $F$11<=$H$11
Not Binding
$F$12
Assembly time
2500.00 $F$12<=$H$12
Binding
Slack
0.00
560.00
0.00
25
Excel Solver Sensitivity Report
Adjustable Cells
Cell
Name
$B$5
Solution value VCR
$C$5
Solution value Stereo
$D$5
Solution value TV
$E$5
Solution value DVD
Final
Reduced
Objective
Allowable
Allowable
Value
Cost
Coefficient
Increase
Decrease
0.00
-1.00
29.00
1.00
1E+30
380.00
0.00
32.00
40.00
1.67
0.00
-8.00
72.00
8.00
1E+30
1060.00
0.00
54.00
10.00
5.00
Constrains
Cell
Name
Final
Shadow
Constraint
Allowable
Allowable
Value
Price
R.H. Side
Increase
Decrease
$F$10
Electronic comp
4700.00
2.00
4700.00
2800.00
950.00
$F$11
Non-electronic comp
3940.00
0.00
4500.00
1E+30
560.00
$F$12
Assembly time
2500.00
24.00
2500.00
466.67
1325.00
26
Excel Solver Sensitivity Report
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) 27
Excel Solver Sensitivity Report
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 $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 $80:
Objective Coefficient – Allocable Decrease ($72 - infinity) and
Objective Coefficient – Allocable Increase ($72 + $8).
28
Excel Solver Sensitivity Report
Constraints
Nonzero Shadow Prices:
• Electronic Components, Shadow price $2
– Each additional unit of electronic components will allow
Anderson to increase its profit by $2.
– The shadow price is $2 for RHS between (3750, 7500).
RHS - Allocable Increase (4700 + 2800) and
RHS - Allocable Decrease (4700 - 950).
29
Excel Solver Sensitivity Report
Constraints
Nonzero Shadow Prices:
• Assembly Time, Shadow price $24
– Each additional hour of assembly time will allow Anderson
to increase its profit by $24.
– The shadow price is $24 for RHS between (1175, 2966.67).
RHS - Allocable Increase (2500 + 466.67) and
RHS - Allocable Decrease (2500 - 1325).
30
Excel Solver Sensitivity Report
Constraints
Zero Shadow Price:
• Non-electronic components, Shadow price $0
– Nonbinding constraint, 560 units of unused resources
– The shadow price is $0 for RHS between (3940, infinite).
RHS - Allocable Increase (infinite) and
RHS - Allocable Decrease (4500 - 560).
31
4.5 Simultaneous Changes Using the 100% Rule
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 units in electronic
component, 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. 32
4.5 Simultaneous Changes In Parameter Values
Anderson Electronics
• 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).
33
4.5 Simultaneous Changes In OFC Values
Anderson Electronics
• 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.
34
4.6 Pricing-out New Variables
• Information given in sensitivity report can be used to
study impact of introduction of new decision
variables (products).
• For example:
– If problem is re-solved with a new product in model,
will it be recommend that a new product be made?
– Or, will it be recommend that a new product not be
made, and continue making same products (that is,
stereos and DVD players)?
35
Could Anderson Electronics Propose a New Product?
Anderson Electronics
• Anderson Electronics considers a new product,
home-theater system (HTS). Could the
company propose this new product?
• Answer to such question involves a procedure
called pricing-out.
36
Pricing-Out Procedure
Home-Theater System (HTS)
• Requires:
– 5 units of electronic components
– 4 units of non-electronic components
– 4 hours of assembly time.
• Selling price: $175 per unit.
• The actual cost is 5 x $7 + 4 x $5 + 4 x 10 = $95.
• The net profit is $175 - $95 = $80.
37
Pricing-out procedure
Home-Theater System (HTS)
• Resources required to make this player:
– No longer available to meet existing production plan (380 stereos and
1060 DVD players) for $69,400 total profit.
• Checking validity of the 100% Rule:
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
• Profit loss if the resources are used for 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.
38
Pricing-out procedure
Home-Theater System (HTS)
• 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.
• The unit profit of HTS is $80. Therefore, Anderson
Electronics should not propose this new product.
39
Revised Excel Layout
Anderson Electronics
V
S
T
T
H
VCR
Stereo
TV
DVD
HTS
Solution value
0.00
380
0.00
1060
0.00
Selling price per unit
$70
$80
$150
$110
$175
$147,000
<-- Revenue
Cost price per unit
$41
$48
$78
$56
$95
$77,600
<-- Cost
Profit
$29
$32
$72
$54
$80
$69,400
<-- Objective
Constraints
Cost
Electronic comp
3
4
4
3
5
4700.00
<=
4700
$7
Non-electronic comp
2
2
4
3
4
3940.00
<=
4500
$5
Assembly time
1
1
3
2
4
2500.00
<=
2500
$10
LHS
Sign
RHS
40
Revised Excel Solver Sensitivity Report
Adjustable Cells
Cell
Name
Final
Reduced
Objective
Allowable
Allowable
Value
Cost
Coefficient
Increase
Decrease
0.00
-1.00
29.00
1.00
1E+30
380.00
0.00
32.00
40.00
1.67
0.00
-8.00
72.00
8.00
1E+30
Solution value DVD
1060.00
0.00
54.00
10.00
5.00
Solution value HTS
0.00
-26.00
80.00
26.00
1E+30
$B$5
Solution value VCR
$C$5
Solution value Stereo
$D$5
Solution value TV
$E$5
$F$5
Constraints
Cell
Name
Final
Shadow
Constraint
Allowable
Allowable
Value
Price
R.H. Side
Increase
Decrease
$G$10
Electronic comp
4700.00
2.00
4700.00
2800.00
950.00
$G$11
Non-electronic comp
3940.00
0.00
4500.00
1E+30
560.00
$G$12
Assembly time
2500.00
24.00
2500.00
466.67
1325.00
41
4.7 Sensitivity Analysis - Minimization
Example
Burn-Off Diet Drink
• Plans to introduce miracle drink that will magically
burn fat away.
Ingredient A
Ingredient B
Ingredient C
Ingredient D
Requirement
Chemical X
3
4
8
10
At least 280
units
Chemical Y
5
3
6
6
At least 200
units
Chemical Z
10
25
20
40
At most 1,050
units
4 cents
7 cents
6 cents
3 cents
Cost per
ounce
42
Burn-Off Diet Drink LP Formulation
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
43
Excel Solution
A
B
C
D
Ingr A
Ingr B
Ingr C
Ingr D
10.250
0.000
4.125
21.625
4
7
6
3
130.625
Daily dosage
1
1
1
1
36.00
>=
36
Chemical X
3
4
8
10
280.00
>=
280
Chemical Y
5
3
6
6
205.75
>=
200
Chemical Z
10
25
20
40
1050.00
<=
1050
LHS
Sign
RHS
Number of ounces
Cost (cents)
<-- Objective
Constraints
44
Solver Answer Report
Burn-Off Diet Drink
Target Cells
Cell
$F$6
Name
Cost (cents)
Original Value
0.000
Final Value
130.625
Adjustable Cells
Cell
Name
Original Value
Final Value
$B$5
Number of ounces Ingr A
0.000
10.250
$C$5
Number of ounces Ingr B
0.000
0.000
$D$5
Number of ounces Ingr C
0.000
4.125
$E$5
Number of ounces Ingr D
0.000
21.625
Constraints
Cell
Name
Cell Value
Formula
Status
Slack
1050.000 $F$11<=$H$11
Binding
0.000
36.000 $F$8>=$H$8
Binding
0.000
Chemical X
280.000 $F$9>=$H$9
Binding
0.000
Chemical Y
205.750 $F$10>=$H$10
Not Binding
5.750
$F$11
Chemical Z
$F$8
Daily dosage
$F$9
$F$10
45
Solver Sensitivity Report
Adjustable Cells
Cell
Name
Final
Reduced
Objective
Allowable
Allowable
Value
Cost
Coefficient
Increase
Decrease
$B$5
Number of ounces Ingr A
10.250
0.000
4.000
3.500
2.500
$C$5
Number of ounces Ingr B
0.000
5.688
7.000
1E+30
5.688
$D$5
Number of ounces Ingr C
4.125
0.000
6.000
15.000
2.333
$E$5
Number of ounces Ingr D
21.625
0.000
3.000
3.800
1E+30
Constraints
Cell
Name
Final
Shadow
Constraint
Allowable
Allowable
Value
Price
R.H. Side
Increase
Decrease
1050.000
-0.238
1050.000
47.143
346.000
36.000
3.750
36.000
16.500
1.278
Chemical X
280.000
0.875
280.000
41.000
11.000
Chemical Y
205.750
0.000
200.000
5.750
1E+30
$F$11
Chemical Z
$F$8
Daily dosage
$F$9
$F$10
46
Change in Objective Function Coefficient (OFC)
Nonzero Reduced Cost: Ingredient B
• The reduced cost of ingredient B is $5.688.
– Each ounce of ingredient B used to make the drink will
cause the total cost per daily dosage to increase by 5.688
cents.
– The current cost of ingredient B is 7 cents. If the cost of
ingredient B is lower by 5.688 cents, then it becomes costeffective to use this ingredient.
– When the cost of ingredient B is above 1.312 cents (=75.688), the current corner point solution remains optimal.
47
Change in Objective Function Coefficient (OFC)
Zero Reduced Cost: Ingredient C
• The reduced cost of ingredient C is $0.
– The current cost of ingredient C is 6 cents per
ounce. The range for the cost coefficient of this
ingredient is between 3.667 cents (=6-2.333) and
21 cents (=6+15).
– When the cost of ingredient C is between this
range, the current corner point solution remains
optimal.
48
Changes in Right-hand-side (RHS)
Nonzero Shadow Price: Chemical X
•
The shadow price of chemical X is 0.875.
•
For each additional unit of chemical X required to
be present in the drink, the total cost will increase
by 0.875 cents.
•
The shadow price remains to be 0.875 if the
requirement for chemical X is between 269 units
(=280-11) and 321 units (=280+41).
49
Changes in Right-hand-side (RHS)
Nonzero Shadow Price: Chemical Z
•
The shadow price of chemical Z is -0.238.
•
Each unit increase in the maximum limit allowed
for chemical Z will reduce the total cost by 0.238
cents.
•
The shadow price remains to be -0.238 if the
maximum limit is between 704 units (=1050-346)
and 1097.143 units (=1050+47.143).
50
Simultaneous Changes In Parameter Values
Burn-Off can decrease the minimum requirement for chemical
X by 5 units provided the maximum limit allowed for
chemical Z is reduced by 50 units.
• The sum of each proportion of change to allowable change
is
5/11 + 50/346 = 0.399 < 1
• Since sum does not exceed 1, information provided in
sensitivity report is valid to analyze impact of changes.
• The reduced cost from the change in chemical X is
0.875 x 5 = 4.375 cents.
• The reduced cost from the change in chemical Z is
0.238 x 50 = 11.9 cents.
• The net impact is an increase in total cost of 7.525 cents
(=11.9-4.375).
51
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.
52