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Chapter 5 - Sensitivity Analysis Using an Applied
Approach - Learning Objectives
- Understand what sensitivity analysis is and be able to
define it, explain why we use it, and justify its
importance.
- Understand how to conduct sensitivity analysis for
the objective function coefficients considering
the range of the objective function coefficients
the reduced cost
the right hand side values of the constraints
considering
the range of the right hand side values
the shadow (or dual) prices of the constraints
- Be able to conduct sensitivity analysis using the
graphical method.
- Be able to conduct sensitivity analysis using LINDO.
You should be able to interpret the entire LINDO output
for a problem.
- Understand the concept of complementary slackness and
what it means.
- Understand parametric analysis and its application.
Sensitivity Analysis
Sensitivity analysis is concerned with examining the effect
that changes in the LP parameters have on the LP's optimal
solution.
These include changes to the objective function
coefficients, and right hand side values.
Why would this be a concern?
For example, let's suppose that in the original formulation
and solution to an LP problem, it was thought that variable
x i
would generate $10 of profit. Well suppose now, that it
only generates $9 of profit. Is the same solution valid?
Does the LP problem need to be solved again?
In another example, suppose that we thought that there
were 1000 hours of machine time. Unexpectedly, a
machine breaks down, reducing the number of hours to 800
hours. Is the same solution valid (as in the original
formulation)? Does the LP problem need to be solved
again?
Overview of Sensitivity Analysis
Example, Section 5.1, pg 201, Problem 5
max z = 3x 1
st (1) x 1
(2) 2x 1
x 1
, x 2
This could be solved using the graphical solution technique
or the simplex method. Let's solve it graphically.
a) For what values of the price of a type 1 radio would the
current basis remain optimal?
This analyzes the effect of a change in an objective
function coefficient.
The slope of any isoprofit line z = -3/
For any isoprofit line the value of the objective function is
constant and equals 3x 1
Replace the '3' with a 'c 1
The slope now becomes - c 1
If the slope of z becomes steeper, point C will no longer be
optimal, but point B will become optimal. This occurs
where:
-c 1
c 1
If the slope of z becomes flatter, point C will no longer be
optimal, but point D will become optimal. This occurs
where:
-c 1
c 1
So, in order for the basis to remain optimal, c 1
, the profit
of a type 1 radio must be:
1 £ c 1
c 1
= price - 5 - 12 - 5
= price - 22
1 £ price - 22 £ 4
23 £ price £ 26
b) For what values of the price of a type 2 radio would the
current basis remain optimal?
This also analyzes the effect of a change in an objective
function coefficient.
The slope of z = -3/
For any isoprofit line the value of the objective function is
constant and equals 3x 1
Replace the '2' with a 'c 2
The slope now becomes - 3/c 2
If the slope of z becomes steeper, point C will no longer be
optimal, but point B will become optimal. This occurs
where:
-3/c 2
c 2
If the slope of z becomes flatter, point C will no longer be
optimal, but point D will become optimal. This occurs
where:
-3/c 2
c 2
d) If laborer 2 were willing to work up to 60 hours per week,
would the current basis remain optimal? Find the new optimal
solution to the LP.
This analyzes the effect of a change in a RHS on the LP's
optimal solution.
e) Find the shadow price of each constraint.
What is a shadow price? Each constraint has a shadow
price. It is the amount by which the optimal z-value is
improved (increased for a max, or decreased for a min) if
the RHS of the constraint is increased by 1. Assuming the
change in the constraint leaves the current basis optimal.
For a two variable problem it is easy to determine the
shadow price.
For this problem, we know that the optimal solution is
occurring at the intersection of the two constraints.
So, for constraint 1, the shadow price is:
x 1
= 40 + D
2x 1
solve simultaneously, yielding
x 1
= 20 - D/
x 2
= 10 + 2 D/
plug into z = 3x 1
yields z = 80 + D/
so, the shadow price for constraint 1 = $1/
In other words, for each additional unit of labor 1,
the profit will increase by $1/3.
LINDO and Sensitivity Analysis
LINDO produces information which is helpful for
sensitivity analysis.
In LINDO type “Yes” when queried after using the GO
command.
Let's follow along in our book on pg 212-3, problem 4,
Gepbab Manufacturing.
Objective Function Coefficient Ranges
LINDO- allowable increase and allowable decrease
This is the increase/decrease with the current basis
remaining optimal. It also assumes that only one variable
is changing.
Note, the value of the decision variables remains
unchanged, however, the value of the objective function
will change.
Example.
Example
RHS Ranges
LINDO- allowable increase and allowable decrease
This is the increase/decrease with the current basis
remaining optimal. It also assumes that only one variable
is changing.
Note, the value of the decision variables may change, and
the value of the objective function will change.
Use of Shadow Prices
The shadow prices can help a manager to answer the question:
What is the maximum amount that I would be willing to pay for
an additional unit of a resource?
Note, if the shadow price is $0, this indicates that the resource is
in excess and we would not be willing to pay any additional
amount for the resource.
Example - Problem 4, Section 5.3, pg 218
What is the most that Gepbab would be willing to pay for
another unit of capacity at plant 1?
