Managerial Decision Making
Chapter 4
Sensitivity Analysis
Introduction

When solving an LP problem we assume that values of all
model coefficients are known with certainty. Such certainty
rarely exists.

Sensitivity analysis helps answer questions about how sensitive
the optimal solution is to changes in various coefficients in a
model.
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MAX (or MIN):
Subject to:
c 1 X1 + c 2 X2 + … + c n Xn
a11X1 + a12X2 + … + a1nXn <= b1
:
ak1X1 + ak2X2 + … + aknXn >= bk
:
am1X1 + am2X2 + … + amnXn = bm
How sensitive is a solution to changes in the ci, aij, and bi?
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Approaches to Sensitivity Analysis
Change the data and re-solve the model!
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Sometimes this is the only practical approach.
Use Solver’s Sensitivity Analysis Report
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Amounts by which objective function coefficients, ci ,can
change without changing the optimal solution.
The impact on the optimal objective function value of
changes in constrained resources (RHS coefficients, bi ).
The impact changes in constraint coefficients, aij, will have on
the optimal solution.

When solving LP problems, be sure to select the “Assume
Linear Model” option in the Solver Options dialog box as
this allows Solver to provide more sensitivity information
than it could otherwise do.
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Recall Blue Ridge Hot Tubs Example...

X1 = number of Aqua-Spa tubs to produce
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X2 = number of Hydro-Lux tubs to produce
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Max
350X1 + 300X2
(profit in dollars)
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Subject to
X1 + X2 <= 200
(pumps)
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9X1 + 6X2 <= 1566
(hours of labor)
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12X1 + 16X2 <= 2880
(feet of tubing)
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X1, X2 >= 0
See file Fig4-1.xls
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Answer Report

Shows the optimal solution and optimal objective value.
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Shows for each constraint:
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Slack: the difference between RHS and LHS calculated for the
optimal solution.
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Binding-Nonbinding Status: a constraint is binding if it
prevents from achieving a better objective value. If binding
constraint is removed or relaxed, the solution may improve.
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A binding constraint will always have slack = 0.
Sensitivity Report: Changes in Objective
Function Coefficients (ci)
“Allowable Increase” and “Allowable Decrease” indicate the amount
by which an objective function coefficient can change without
changing the optimal solution, assuming all other coefficients remain
constant.
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Adjustable Cells
Cell
Name
$B$5 Number to make Aqua-Spas
$C$5 Number to make Hydro-Luxes
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Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
122
0
350
100
50
78
0
300
50 66.66667
If the unit profit for Aqua-Spas stays between 350 – 50 and 350 + 100,
then the optimal solution does not change.
If the unit profit for Aqua-Spas is decreased by more than 50 or
increased by more than 100, then the optimal solution may change.
We only know what will happen if only one coefficient is changed.
Values of zero (0) in the “Allowable Increase” or “Allowable Decrease”
indicate that an alternate optimal solution may exist.
Even if the optimal solution remains the same, the optimal objective
value may change.
How Changes in Objective Coefficients
Change the Slope of the Level Curve
X2
250
original level curve
200
new optimal solution
150
original optimal solution
100
new level curve
50
0
0
50
100
150
200
250
X1
Sensitivity Report: Changes in
Constraint RHS Values
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The “Shadow Price” of a constraint indicates the amount
by which the objective function value increases
(decreases) given a unit increase (decrease) in the RHS
value of the constraint, assuming all other coefficients
remain constant.
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Shadow Prices hold only for RHS changes falling within
the values in “Allowable Increase” and “Allowable
Decrease” columns.
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Shadow prices for nonbinding constraints are always zero.
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Sensitivity Report: Changes in
Constraint RHS Values continued
Constraints
Cell
$D$9
$D$10
$D$11
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Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side
Increase Decrease
200
200
200
7
26
1566
16.67
1566
234
126
2712
0
2880
1E+30
168
If we have 5 extra pumps, the profit will increase by 5 * “Shadow Price”
= 5 * $200 = $1000.
If we find that 10 pumps are broken, the profit will decrease by $2000.
If 50 extra pumps are ordered, we cannot tell by how much the profit
will increase without resolving the problem. However, we know that it
will increase by at least “Allowable increase” * “Shadow Price” = $1400.
When a RHS of a binding constraint is changed, we do not know what
happens to the optimal decision, only what happens to the optimal
objective function.
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Name
Pumps Req'd Used
Labor Req'd Used
Tubing Req'd Used
However, if changing the RHS changes the optimal objective value, then the
optimal solution must also be changing. To find it, the problem should be resolved.
How Changing an RHS Value Can Change
the Feasible Region and Optimal Solution
X2
250
Suppose available labor hours
increase from 1,566 to 1,728.
200
150
old optimal solution
old labor constraint
100
new optimal solution
50
new labor constraint
0
0
50
100
150
200
250
X1
Add a Decision Variable
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Suppose a new hot tub model, the Typhoon-Lagoon, is being
considered.
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Its unit profit is $320 and it requires:
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1 pump (Shadow Price = $200)
8 hours of labor (Shadow Price = $16.67)
13 feet of tubing (Shadow Price = $0)
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Q: Would it be profitable to produce any Typhoon-Lagoons? In
other words, would it be profitable to shift some resources from
Aqua-Spas and Hydro-Luxes to produce Typhoon-Lagoons?
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See file Fig4-8.xls
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Other Uses of Shadow Prices

For each resource constraint, its Shadow Price represents the
marginal value of the resource.
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Suppose a new hot tub model, the Typhoon-Lagoon, is being
considered. Its unit profit is $320 and it requires:
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Q: Would it be profitable to produce any Typhoon-Lagoons? In
other words, would it be profitable to shift some resources from
Aqua-Spas and Hydro-Luxes to produce Typhoon-Lagoons?
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1 pump (Shadow Price = $200)
8 hours of labor (Shadow Price = $16.67)
13 feet of tubing (Shadow Price = $0)
A: The marginal profit for one Typhoon Lagoon equals to (unit profit) –
(cost of resources) = $320 – ($200*1 + $16.67*8 + $0*13) = – $13.33.
Since the value is negative, the answer is: No!
Marginal profit is calculated assuming that all Shadow Prices can be
used at the same time when all RHS changes are “small enough”.
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Other Uses of Shadow Prices continued
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Q: Suppose a Typhoon-Lagoon required only 7 labor
hours rather than 8. Is it now profitable to produce any?
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Q: What is the maximum amount of labor TyphoonLagoons could require and still be profitable?
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A: $320 – ($200*1 + $16.67*7 + $0*13) = $3.31 = Yes!
To find out how many Typhoon-Lagoons will be produced, the
model with three products should be solved.
A: We need $320 – ($200*1 + $16.67*L + $0*13) >= 0
The above is true if L <= $120/$16.67 = 7.20
See file Fig4-8.xls
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The Meaning of Reduced Costs

The Reduced Cost for each product equals its per-unit marginal value when
resources are priced at their Shadow Prices.
Adjustable Cells
Cell
Name
$B$5 Number to make Aqua-Spas
$C$5 Number to make Hydro-Luxes
$D$5 Number to make Typhoon-Lagoones
Final
Value
122
78
0
Reduced
Cost
Objective Allowable Allowable
Coefficient Increase Decrease
0
350
100
20
0
300
50
40
-13.33
320
13.33
1E+30
RC of Aqua-Spa =
$350 – ($200*1 + $16.67*9 + $0*12) = $0
RC of Hydro-Lux =
$300 – ($200*1 + $16.67*6 + $0*16) = $0
RC of Typhoon-Lagoon = $320 – ($200*1 + $16.67*8 + $0*13) = – $13.33
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At optimality, if a variable is not on its lower (upper) bound, the reduced
cost is zero.
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If a variable is on its lower (upper) bound, the reduced cost tells us by how
much the objective function value would change per unit change of the
lower (upper) bound.
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