Download Sensitivity Analysis with Excel

Sensitivity Analysis with Excel
1
Lecture Outline
Sensitivity Analysis
Effects on the Objective Function Value (OFV):
• Changing the Values of Decision Variables
• Looking at the Variation in OFV:
Excel One- and Two-Way Data Tables, and Scenario Manager
• Finding the Optimum OFV: Excel Solver
Effects on the Optimum:
• Making One Change to the Model Parameters:
• Changing a Decision Variable Coefficient in a Constraint
(Graphical intuitions)
• Changing the Right Hand Side (RHS) of a Constraint:
Slack & Shadow Price
• Changing a Decision Variable Coefficient in the Objective Function:
Reduced Cost
• Making Multiple Changes to the Model Parameters
• The 100% Rule
• SolverTable
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1
Steps in Modeling
STEP 1: Formulate
STEP 2: Solve (Find the Optimum)
STEP 3: Do a Sensitivity Analysis
3
Problem Statement
Boxers and Briefs Example - Revisited
Champion Sports manufactures two types of custom men's
underwear: boxers and briefs. How many boxers and how
many briefs should be produced per week, to maximize
profits, given the following constraints…
– The (profit) contribution per boxer is $3.00, compared to $4.50
per brief.
– Briefs use 0.5 yards of material; boxers use 0.4 yards. 300
yards of material are available.
– It requires 1 hour to manufacture one pair of boxers and 2
hours for one pair of briefs. 900 labors hours are available.
– There is unlimited demand for boxers but total demand for
briefs is 375 units per week.
– Each boxer uses 1 insignia logo and 600 insignia logos are in
stock.
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2
Algebraic
Formulation
5
The Algebraic LP Formulation
Revisited
•
Variables: number of Boxers, number of Briefs.
•
Objective function:
–
•
Objective Function Coefficients
maximize ( $3.00 x Boxers ) + ( $4.50 x Briefs )
Constraints:
Constraint Coefficients
–
Material: ( .4 x Boxers ) + ( .5 x Briefs ) <= 300 yards
–
Labor:
–
Demand: ( 0 x Boxers ) + ( 1 x Briefs ) <= 375 units
–
Logos:
–
Non-Negativity:
Boxers >= 0
Briefs >= 0
( 1 x Boxers ) + ( 2 x Briefs ) <= 900 hrs
( 1 x Boxers ) + ( 0 x Briefs ) <= 600
Constraints Right Hand Sides (RHS)
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3
Graphical
Formulation
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The Graphical LP Formulation - Revisited
Optimal Solution
900
Hours
800
700
Boxers
Logos
600
500
F
E
D
Optimum
400
300
Demand
Total
Profit
$1,800.00
$1,687.50
$2,137.50
$2,400.00
$2,340.00
$1,800.00
A
200
100
A
B
C
D
E
F
$3.00 $4.50
Boxers Briefs
300
200
0
375
150
375
500
200
600
120
600
0
Material
C
B
100 200 300 400 500 600 700 800 900
Briefs
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4
Excel
Formulation
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Excel Formulation - Revisited
Boxers and Briefs Example: Profit Maximization
See boxers_and_briefs_example.xls in the downloads for today’s class.
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5
Excel Tools for Sensitivity Analysis
• One-way Data Table, Two-Way Table, Scenario Manager
• Solver Answer Report
Pertains to changing RHS of
a constraint
• Slack
• Solver Sensitivity Report
• Shadow Price, Allowable Increase / Decrease
• Reduced Cost, Allowable Increase / Decrease
• SolverTable
Pertains to changing
Objective Function
Coefficients
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Changing the Value of Decision Variables
With Two- Way Data Tables
The two-way data table below shows the effect, on profit, of
changing the two decision variables (i.e. the number of boxers produced, and
the number of briefs produced). This initial analysis assumes there are no
constraints. Clearly, if constraints are taken into account, the maximum profit in
the data table shown below ($4350, for 700 boxers and 500 briefs) would
change since that is not a feasible solution.
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6
Steps in Modeling
STEP 1: Formulate
STEP 2: Solve (Find the Optimum)
STEP 3: Do a Sensitivity Analysis
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Changing the Value of Decision Variables
With Two- Way Data Tables
The two-way data table below shows the effect on profit of changing the number
of boxers and briefs produced. Conditional formatting has been used to highlight
infeasible solutions (in red) and the optimal feasible solution (in green).
Don’t panic ! You aren’t expected to know how to construct the table /
formula shown below – its only shown to illustrate how complex it is to
create a table that shows the feasible and optimal solutions ! Notice
that we were lucky in this instance that the optimum solution falls at a
point that was in our table: had we chosen broader intervals for
number of boxers and briefs we would have found a good, but not
optimal, solution.
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Excel Answer Report - Revisited
Boxers and Briefs Example: Profit Maximization
Fortunately, as we’ve seen Excel
provides us with an easy mean of
finding the optimal feasible
solution: simply use Excel Solver.
– Make sure Solver is
available:
• If not, Tools | Add-Ins… |
Solver Add-in
– Use it:
• Tools | Solver…
– Read the Answer Report to
find the optimal value, and the
decision variable settings at
this optimum.
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Excel Answer Report
Optimum
Objective
Function Value
Optimum
Product Mix
Status of the
Constraints
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8
Steps in Modeling
STEP 1: Formulate
STEP 2: Solve (Find the Optimum)
STEP 3: Do a Sensitivity Analysis
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Answer Report
The Answer Report also shows us which constraints are binding and nonbinding at the optimum. “Slack” indicates the spare capacity on a non-binding
constraint at the optimum.
– Slack = 0 implies constraint is binding: resources are exhausted
– Slack ≠ 0 implies constraint is non-binding: there are left-over resources
We can see below, for instance, that:
– getting additional logos
wouldn’t help us improve profit
since we already have 100
unused logos at the optimum
– advertising our briefs to
stimulate demand (at the
current price) would be
money wasted, since demand
for briefs is already greater
than the number of briefs we
should produce at the
optimum.
– material and labor constraints
are cramping our profit.
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9
Slack and Binding
• Slack measures unused available resources
• Binding constraints
– Optimal solution lies on binding constraints
– Multiple binding constraints means solution is at intersection of
constraints: a vertex
• Slack and binding constraints
– Slack = 0 : the resource is exhausted, the constraint is binding,
the optimal vertex includes this constraint.
– Slack ≠ 0 : spare capacity is available, constraint is not binding
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Excel Sensitivity Analysis Report
Choose Sensitivity to see a more detailed Sensitivity Report...
Pertain to
Objective
Function
Coefficient
Ranging
Pertain to
Right Hand Side
(RHS) Ranging
on Constraints
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10
Excel Sensitivity Analysis Report
The Excel Sensitivity Analysis Report allows you to see:
• … over what range and under what conditions the
components of a solution remain unchanged
• … how sensitive a solution is to changes in the data, and
to get an insight into how technological improvements
may affect optimum values.
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Sensitivity Analysis
Changing a Decision Variable
Co-Efficient in a Constraint
Graphical Intuitions
The Sensitivity Report doesn’t tell us anything about what happens when the
coefficients of a decision variable in a constraint change, but here are some
graphical intuitions…
800
Boxers
700
600
500
400
300
Hours
Assume we
changed the amount
of material required
per brief from 0.5
Logos
yards to 1 yard.
Notice how the
shape of the
Demand
feasible region
changes, and the
optimal solution
changes.
200
Material
100
900
700
Briefs
Logos
600
500
Demand
400
300
200
100
100 200 300 400 500 600
Hours
800
Boxers
900
Material
100 200 300 400 500 600
Briefs
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11
Sensitivity Analysis Report
Changing the Right Hand Side (RHS)
of a Constraint
Graphical Intuitions
Changing the RHS of a constraint causes the constraint line to shift left or right,
but does not alter the slope of the line !
800
Boxers
700
600
500
400
300
200
100
Hours
Assume an extra
Demand
100 yards of
material became
Logos available (increasing
our total available to
400 yards). Notice
how the material
constraint shifts
right. The change is
greater than the
Material ‘allowable increase’
(see later) and the
optimum solution
vertex changes.
100 200 300 400 500 600
900
Hours
Demand
800
700
Logos
600
Boxers
900
Material
500
400
300
200
100
100 200 300 400 500 600
Briefs
Briefs
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Sensitivity Analysis
Changing the Right Hand Side (RHS)
of a Constraint
The Shadow Price
What is meant by the optimal vertex ?
The optimal vertex is :
“what intersection of constraints is the optimal solution to be found at?”.
From the previous slide, we saw that the original optimal vertex was:
“at the intersection of the materials and hours constraints”.
However, when we increased the RHS of the materials constraint beyond the
maximum ‘allowable increase’ (i.e. by 100 yards), the optimal vertex shifts to:
“at the intersection of the logos and hours constraints”.
Had the change in materials been within the allowable increase, then the
optimal vertex would have stayed the same (i.e. it would still have been “at the
intersection of the materials and hours constraints) but the optimal product
mix and the optimal solution value would have changed.
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12
Sensitivity Analysis
Changing the Right Hand Side (RHS)
of a Constraint
Graphical Intuitions
Changing the RHS of a constraint causes the constraint line to shift to a
parallel position to the left or right, but does not alter the slope of the line !
However, the change is less than
the ‘allowable increase’ for this
constraint and the optimum
solution vertex stays the same:
the optimal solution is still at the
intersection of the materials and
labor hours constraints .
900
Hours
Demand
800
700
Logos
600
Boxers
Assume a extra 100 hours of labor
became available (increasing our
total available to 1000 hours).
Notice how the labor constraint
shifts right, and the optimal
solution value and optimal
product mix change.
500
400
300
Material
200
100
100 200 300 400 500 600
Briefs
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Sensitivity Analysis
Changing the Right Hand Side (RHS)
of a Constraint
•
Tighten constraint: make feasible region smaller.
– Optimal value can only get worse (fewer choices)
•
Loosen (relax) constraint: make feasible region larger.
– Optimal value can only do better (more choices)
•
Assume b is a positive number and the constraint line has the form
y = ax + b (i.e. y – ax = b) then increasing the RHS (i.e. b) will cause the
line to shift vertically up and will:
– Tighten the constraint if it’s a lower-bound constraint
– Loosen the constraint if its an upper-bound constraint
So notice that increasing the RHS may have positive or negative effects
on the optimum, depending on the type of constraint !
•
In contrast, if b is a positive number and the constraint line has the form
y = ax – b (i.e. ax – y = b) then increasing the RHS (i.e. b) will cause the
line to shift vertically down !
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13
Sensitivity Analysis Report
Changing the Right Hand Side (RHS)
of a Constraint
The Shadow Price
The Shadow Price for a constraint is the change in the optimal objective
function value per unit increase in the Right Hand Side (RHS) of a given
constraint.
The Shadow Price only remains valid within the Allowable Increase and
Decrease shown for that Shadow Price.
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Shadow Prices
Example
A stain is found on 15 yards of material, reducing material from
300 to 285 yards. How does this affect optimal profit ?
New optimal profit
= Old optimal profit - (shadow price x yards)
= $2,400 – ($5/yard x 15 yards) = $2,325
Notice the Allowable range for the Materials Usage constraint.
What could you say about a stain on 60 yards?
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14
Shadow Prices
Example
Labor is willing to negotiate 100 additional hours of production work.
How much of a premium should management pay for overtime hours?
Addition to optimal profit = shadow price x hours
= $1/hour x 100 hours = $100
If $9.28 for 900th hour, then $10.28 for 901st hour. But remember, the
productivity rate will change when people work longer hours…
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Increase Labor by 100 hours
900
Hours
800
Boxers
700
600
500
Logos
F
E
Demand
D
400
300
200
G
A
$3.00 $4.50
Total
Boxers Briefs Contribution
A 300 200 $1,800.00
B
0 375 $1,687.50
C
150 375 $2,137.50
D 500 200 $2,400.00
E
600 120 $2,340.00
F
600
0 $1,800.00
G 333.3 333.3 $2,500.00
C
100
Material
B
100 200 300 400 500 600 700 800 900
Briefs
30
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Shadow Prices
Example
How much would you pay for one additional insignia logo?
Nothing: logos are not constraining the solution!
Notice the range for logos: what is 1E+30?
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Shadow Price / RHS Ranging
• Shadow price
– Shadow price = marginal change to objective function value of
increasing constraint RHS by 1 unit
– Scaling issues: changing one model unit
(e.g. model may be in millions of units …)
• Slack and shadow price
– Shadow price = 0 implies constraint is not binding
– Shadow price ≠ 0 implies Slack = 0
• Slack and allowable increase/decrease
– How much you must tighten a constraint to make it bind
– Notice that, for a non-binding constraint, either the allowable
increase or the allowable decrease will be equal to the slack,
since either adding or subtracting the slack to / from the RHS
will make the constraint bind.
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16
Shadow Price / RHS Ranging
• Shadow price on a constraint:
– Change in optimal objective function value per unit change in righthand-side of the constraint
– zero if constraint is non-binding
• Shadow price and RHS ranging:
– Allowable increase / decrease =
Range of RHS coefficients for which shadow prices remain valid.
• Optimal value (production mix) will change:
–
–
–
–
If binding constraints are moved, the optimal mix changes
The objective function value changes
The shadow price allows us to predict new optimal value.
Need to resolve the model to get the new mix (decision variables).
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Allowable Increase / Decrease
for the RHS of a Constraint
Allowable increase/decrease in constraints:
• How much you can tighten or relax a constraint RHS and remain
binding (or non-binding) (see also: Slack)
• Range in RHS coefficients for which the shadow price remains valid
(see also: Shadow Price)
• Feasible Region: How much you can reshape the feasible region
without changing the optimal vertex. (The optimal mix will always
change, but the optimal vertex stays the same within the allowable
increase/decrease.)
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Allowable Increase / Decrease
for the RHS of a Constraint
Change within allowable increase/decrease
• Optimal vertex (intersection of the binding constraints)
is unchanged.
• Optimal value of objective function is updated by
Shadow Price.
• Optimal product mix (value of decision variables) is
calculated by re-testing the model.
Change outside allowable increase/decrease
• Start over ….
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Shadow Price, Slack, and Binding
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18
Shadow Price in
Maximization vs Minimization Problems
– For maximization problems:
• increase in the objective function value is good
(+ve shadow price is good, and helps the optimum solution)
• decrease in the objective function value is bad
(-ve shadow price is bad, and hurts the optimum solution)
– For minimization problems:
• increase in the objective function value is bad
(+ve shadow price is bad, and hurts the optimum solution)
• decrease in the objective function value is good
(-ve shadow price is good, and helps the optimum solution)
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Shadow Prices in a Minimization Problem
Our objective in the Big Mac Attack Problem is to achieve least-cost
in meeting our Recommended Daily Allowance (RDA) requirements:
including both upper and lower limits.
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19
Shadow Prices in a Minimization Problem
Sodium was an upper bound constraint. It has a negative shadow price:
increasing the RHS of the sodium constraint would loosen the constraint
and allow us to achieve a better optimum: i.e. a lower meal cost.
Vitamin C was an lower bound constraint. It has a positive shadow
price: increasing the RHS of the Vitamin C constraint would tighten the
constraint and force us to a worse optimum: i.e. a higher meal cost.
Constraints
Cell
$D$21
$E$21
$F$21
$G$21
$H$21
$I$21
$J$21
$K$21
$L$21
$M$21
Name
TOTAL Protein
TOTAL Fat
TOTAL Sodium
TOTAL VitaminA
TOTAL VitaminC
TOTAL VitaminB1
TOTAL VitaminB2
TOTAL Niacin
TOTAL Calcium
TOTAL Iron
Final
Value
80.7
52.5
3000
100
100
138
116
124
100
100
Shadow
Constraint
Price
R.H. Side
0.00000E+00
55
0.00000E+00
54.7
-2.98066E-04
3000
1.57422E-02
100
6.80000E-03
100
0.00000E+00
100
0.00000E+00
100
0.00000E+00
100
1.38042E-02
100
3.38561E-02
100
Allowable
Increase
25.70794559
1E+30
80.04881803
79.62482802
1E+30
38.48083748
16.44004464
24.04707449
78.48290598
28.82593
Allowable
Decrease
1E+30
2.242552634
864.3523347
42.4869867
38.07812436
1E+30
1E+30
1E+30
13.19677205
8.011547469
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Sensitivity Analysis Report
Changing a Decision Variable
Co-Efficient in the Objective Function
The allowable increase and decrease for the Adjustable Cells
(i.e. for the objective function coefficients) tells us how much the objective
function coefficients can change before the optimal solution vertex changes.
Note that the optimal solution value changes as the objective function
coefficients change, but the optimal product mix and vertex stays the same
within the objective function coefficient’s allowable increase/decrease range.
40
20
Sensitivity Analysis Report
Changing an Objective Function Coefficient
Graphical Intuitions
900
Hours
Demand
800
700
Boxers
• Changing an objective function coefficient
changes the slope of the objective function.
• Optimal solution value always changes but,
within the allowable increase / decrease, the
optimal product mix will not: within the
allowable increase decrease the isoprofit line
just swivels around a single point!
• If the slope of the objective function changes
beyond the allowable increase / decrease then
the optimal vertex will change, and the optimal
product mix will change, as illustrated to the
right. At right we see the relative contribution
of briefs (i.e. the coefficient of ‘briefs’ in the
objective function) increasing, causing the
objective function to become steeper, until
eventually (beyond the allowable increase) the
optimal product mix and optimal vertex shift.
The new optimal solution favors more briefs in
the optimal product mix.
Logos
600
500
400
300
Material
200
100
100 200 300 400 500 600
Briefs
41
Sensitivity Analysis Report
Changing an Objective Function Coefficient
Graphical Intuitions
Notice that for an isoprofit line with multiple
optima, the allowable increase and decrease
are zero, since it is impossible for the line to
swivel around a single point and any change
in the objective function coefficients will
cause the optimal vertex to shift !
900
800
Hours
Demand
700
Boxers
Multiple Optima
Notice that, for a certain combination of
objective function coefficients, the objective
function can becomes tangent to a segment
of a constraint line (in this case, to a segment
of the Hours constraint). In this case, every
point along the tangential line segment is an
optimum, so multiple optimal product mixes
are available, all with the same optimal
solution value.
Logos
600
500
400
300
Material
200
100
100 200 300 400 500 600
Briefs
42
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Summary
Changing a Decision Variable
Co-Efficient in the Objective Function
•
A coefficient is associated with each decision variable
•
The allowable increase / decrease for each coefficient is the range
over which the coefficient can vary without changing the product
mix (i.e. without changing the vertex at which the optimal solution
is found)
•
The following do change:
– Objective Function Value
– Shadow Prices
– Reduced Costs
•
Need to resolve the model to find this information
•
Can use to understand flexibility in relative pricing of a product.
43
Summary
Changing a Decision Variable
Co-Efficient in the Objective Function
• Changes “slope” of isoprofit / isocost curve (in 2D)
• Changes decision variable contribution to objective
function
• Does NOT change shape of feasible region
• In 2-dimensional case, the slope of the objective
function changes, but, within the allowable increase /
decrease range, the optimal solution still resides at
the same extreme vertex of the feasible region
44
22
Changing a Decision Variable
Co-Efficient in the Objective Function
Example
A management consultant offers to improve efficiency in the production of
boxers. This would increase the contribution by $0.50 to $3.50. What is the
new mix? What is the increase in weekly profit ?
No change in the product mix.
Change in weekly profit = $0.50 x 500 boxers/week = $250 ($2,650 total)
Note the range: What could you say about $1 increase?
45
Changing a Decision Variable
Co-Efficient in the Objective Function
Example
The contribution of briefs decreases by $0.75 to $3.75. What is
the new mix? What is the decrease in weekly profit?
No change in the product mix.
Change in weekly profit =
$0.75 x 200 briefs/week = $150 ($2,250 total)
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23
Reduced Cost
• Associated with each decision variable.
• Amount by which profit contribution of variable must
be improved before the variable will have a positive
value in the solution.
• Or, rate at which the objective function value will
deteriorate if a variable currently at zero is forced to
increase by a small amount.
• Zero if the variable already appears in the optimal
solution.
47
Reduced Cost (RC)
•
Unit cost (penalty or loss) in optimal objective function value of
forcibly including a Decision Variable (DV) not in the optimal
solution.
•
Necessary change in DV coefficient (‘reduction’ in price) so the
DV is part of the optimal solution.
•
Rate at which the optimal objective function value deteriorates
when a non-optimal DV is required.
•
Reduced cost of a DV happens to be equivalent to the shadow
price of the non-negativity constraint for that DV. Why ?
Because forcing a variable into a solution is the same as
increasing the RHS of its non-negativity constraint (e.g. from ≥0
to ≥1).
•
RC = 0 implies DV is part of the optimal solution.
•
RC ≠ 0 implies DV does not contribute to the optimal objective
function value, and that forcing that DV into the solution would
worsen the optimal solution.
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24
Reduced Costs
Example
Adjustable Cells
Cell
Name
$B$11 Production Boxers
$C$11 Production Padded
$D$11 Production Briefs
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
500
$0.00
3
0.6
0.3
0
($1.00)
6
1
1E+30
200
$0.00
4.5
1.5
0.75
New line: padded briefs, 1 yard of material and 2 hours of
labor. Contribution is $6.00 per padded brief.
Forced to produce one unit of padded brief per week, what
would be cost?
Reduced cost is -$1.00 per padded brief.
49
Amended Model
To Demonstrate Reduced Cost
Force the model to construct ONLY boxers:
• Objective function:
– Maximize ( $10.00 x Boxers ) + ( $1 x Briefs )
• Constraints:
–
–
–
–
–
Material: ( 1 x Boxers ) + ( 0.5 x Briefs ) ≤ 300 yards
Logos: ( 1 x Boxers ) + ( 0 x Briefs ) ≤ 600 logos
Labor:
( 1 x Boxers ) + ( 2 x Briefs ) ≤ 900 hrs
Demand: ( 0 x Boxers ) + ( 1 x Briefs ) ≤ 375 units
Boxers ≥ 0
Briefs ≥ 0
50
25
Reduced Cost
Example
The optimal solution to the amended model is:
300 Boxers, 0 Briefs for Optimal Value: $3,000
Adjustable Cells
Final
Reduced
Objective
Allowable
Allowable
Cell
Name
Value
Cost
Coefficient
Increase
Decrease
$C$3 Decision Variables Boxers
300
0
10
1E+30 7.999999955
$D$3 Decision Variables Briefs
0 -3.999999967 1.00000002 3.999999967
1E+30
Constraints
Cell
$E$6
$E$7
$E$8
$E$9
Name
Material
Logos
Labor
Demand
Final
Value
300
300
300
0
Shadow
Price
10
0
0
0
Constraint
R.H. Side
300
600
900
375
Allowable
Increase
300
1E+30
1E+30
1E+30
Allowable
Decrease
300
300
600
375
Now, the President comes to visit Penn, but he has forgotten his
briefs … so we must manufacture at least one pair. What is the
penalty? … Think carefully …
51
Reduced Cost
Example (Amended Model)
– Maximize: ( $10.00 x Boxers ) + ( $1 x Briefs )
– Material: ( 1 x Boxers ) + ( 0.5 x Briefs ) ≤ 300 yards
Adjustable Cells
Final
Reduced
Objective
Allowable
Allowable
Cell
Name
Value
Cost
Coefficient
Increase
Decrease
$C$3 Decision Variables Boxers
300
0
10
1E+30 7.999999955
$D$3 Decision Variables Briefs
0 -3.999999967 1.00000002 3.999999967
1E+30
Constraints
Cell
$E$6
$E$7
$E$8
$E$9
Name
Material
Logos
Labor
Demand
Final
Value
300
300
300
0
Shadow
Price
10
0
0
0
Constraint
Allowable
Allowable
Only
Binding
Constraint
R.H. Side
Increase
Decrease
300
300
300
600
1E+30
300
900
1E+30
600
375
1E+30
375
Every boxer contributes $10.00 Every brief contributes $1.00 If
we make 1 brief we get $1 more profit, but we lose 0.5 yards of
material, which costs us 0.5 boxers (i.e. $5 of boxer profits)
Thus the marginal loss is $4 (= $1 - $5). Notice that the labor,
logo, and demand constraints weren’t binding so producing an
extra pair of briefs doesn’t cost us anything there !
52
26
Multiple Optima
• We saw earlier that one indicator of multiple optima
was a zero allowable increase and decrease, since
that implied that the objective function overlapped
with a line segment (rather than merely being tangent
to a single point), and thus could not pivot within a
range around a single point.
• As second indicator of multiple optima is finding a
decision variable with a Final Value (at the optimum)
of zero, and a Reduced Cost of zero. This means
that the variable can be forced into the optimal
solution at no cost, and therefore an alternative
optimum is available.
53
Reduced Cost in
Maximization vs Minimization Problems
• Reduced cost tells you the effect on the objective function value of forcing
a variable into the optimal solution. Forcing a variable into the optimal
solution always worsens the solution, irrespective of whether its a max or
min problem. However:
• a worse solution in a max problem involves a lower optimal value
(i.e. negative reduced cost)
• a worse solution in a min problem involves a higher optimal value
(i.e. positive reduced cost).
• This is why:
• max problems have negative reduced costs
• min problems have positive reduced costs
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Changing Multiple Parameters Simultaneously:
The 100% Rule
• We've only changed on parameter at a time. What happens if
we change more than one?
• Use the 100% rule for simultaneous changes in constraint RHSs
and Decision Variable coefficients.
• Calculate each change as a percentage (%) of its respective
allowable increase/decrease.
• If the accumulated (absolute value) % changes are less than
100%, then you sum their shadow price / reduced cost impacts.
• Otherwise, you must recalculate the LP…
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The 100% Rule
•
100% also holds for objective coefficients.
•
Can also be combined for changes in constraint functions.
•
If a single value is outside of range, or if the sum of ratios >
1, then you need to re-compute a solution (resolve the LP)
for new constraints.
•
Remember:
– Change constraint functions, production mix changes
– Change objective function (within allowable increase /
decrease), production mix does not.
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The 100% Rule
Example
If material decreases from 300 to 290 yards and labor increases from 900 to 1000
hours, what is the change in weekly contribution?
100 / 131.25 labor hours + 10 / 52.5 material
= 0.7619 + 0.1905 = 0.9524 < 1.0000
So, change in Objective Function Value =
($1/hr x 100 hrs) - ($5/yard x 10 yards) = $50
So, new Objective Function Value =
$2,450 (= $2,400 + $50)
NOTE: The solution changes to 266.67 boxers and 366.67 briefs
(need to resolve to get this information)
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100% Rule
Example: Pricing out a New Product
Constraints
Cell
$D$4
$D$5
$D$6
$D$7
Name
Yards_Material
Logos
Hours_Labor
Demand_Briefs
Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side Increase Decrease
300
5
300
15
52.5
500
0
600
1E+30
100
900
1
900
131.25
60
200
0
375
1E+30
175
Product designers offer a new line of “padded” briefs that require
1 yard of material and 2 hours of labor. Contribution would be
$6.00 per brief. Should management introduce this line?
Percentage changes: 1 / 52.5 material 2 / 60 labor hours
= 0.019 + 0.033 = 0.052 < 1.0000
Loss from one “padded” brief = reducing relevant constraints
($5.00/yard x 1 yard) + ($1.00/hour x 2 hours) = $7.00
Do not produce! $7.00 cost exceeds the $6.00 contribution
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SolverTable
SolverTable is an Excel Add-In that allows you to produce 2-way data tables
which look at the sensitivity of the optimum solution to changes in any 2
parameters.
Question:
How does SolverTable differ from a regular 2-way data table ?
Answer:
They’re pretty much the same, except:
• SolverTable reruns Excel Solver for each combination of parameter
values, whereas a regular 2-way data table could not do that.
• SolverTable does not auto-update – it merely pastes values. So
SolverTable would need to be rerun if other model parameters
(besides the 2 you are testing) change. In contrast, a regular 2-way
data table uses formulas, and the values of these formulas
automatically update as model parameters change.
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SolverTable
– Getting it:
• Download it from
http://highered.mcgraw-hill.com/sites/0072493682/student_view0/cd_update__solver_table.html
– Installing It:
• Follow the instruction on the web-page above.
• Then open Excel and go to Tools | Add-Ins… | Solver Table Add-in
– Using it:
• Lay out your 2-way table, putting the formula to evaluate under the
different scenarios in the top left hand corner, like you would in a
regular 2-way data table.
• Go to Tools | SolverTable…
Warning: Because of the complexity of the 2-way data tables and SolverTables in the Boxers and
Briefs example spreadsheet, it could take you up to 20 minutes to run the scenario analysis. Press
Escape repeatedly at any time, or Ctrl+Break, if you wish to terminate the analysis.
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SolverTable Example
Changing Multiple Decision Variables Coefficients in a Constraint
The example below shows the results of running SolverTable to investigate the effects
of changing per-unit labor requirements for boxers and briefs on the optimal solution. In
other words, it shows the effect of changing the coefficients of the decision variables in
the labor constraint. The top table shows the effect on the optimal solution value (i.e.
optimal profit in dollars). The bottom table shows the effect on the optimal product
mix. You can see that increasing the number of hours required per product type
(boxers or brief) decreases the amount of that product type in the optimal mix.
Effect on
Optimal
Solution Value
Effect on
Optimal Product
Mix
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Key Points
Constraint Right Hand Sides:
• Slack and Shadow Price
• Changes within the allowable ↑ / ↓ never alter whether a
constraint is (non) binding.
• Change constraint RHS: value of decision variables (product
mix) changes.
Objective Function Coefficients:
• Reduced Cost
• Changes within the allowable ↑ / ↓ never alter the optimal
vertex.
• Changes within the allowable ↑ / ↓: value of decision
variables (product mix) does not change.
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Sensitivity Analysis Report
Summary
Optimal product mix
(Optimal decision
variable values)
Non-zero if the
variable is not
in the solution.
Zero if it is.
Usage of resource
(Left Hand Side of constraint)
Allowable objective function
coefficient range:
Solution vertex stays the same.
Allowable constraint RHS range:
Shadow price is valid.
Increase in optimal objective function value per unit increase in
right hand side (RHS) of constraint.
∆Z = (shadow price) × (∆
∆RHS)
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Sensitivity Analysis
Why Should You Care ?
• Uncertainty – welcome to the real world.
• Slack – what to do with unused resources?
• Iteration – constraints are rarely fixed
e.g. begin with a budget allocation and then evaluate alternatives
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