LINEAR PROGRAMMING
Introduction to Sensitivity
Analysis
Professor Ahmadi
Slide 1
Learning Objectives
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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.
Slide 2
Sensitivity Analysis
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Sensitivity analysis (or post-optimality analysis) is
used to determine how the optimal solution is
affected by changes, within specified ranges, in:
• the objective function coefficients
• the right-hand side (RHS) values
Sensitivity analysis is important to the manager who
must operate in a dynamic environment with
imprecise estimates of the coefficients.
Sensitivity analysis allows the manager to ask certain
what-if questions about the problem.
Slide 3
Range of Optimality: The Objective Function
Coefficients
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A range of optimality of an objective function
coefficient is found by determining an interval for
the coefficient in which the original optimal
solution remains optimal while keeping all other
data of the problem constant. (The value of the
objective function may change in this range.)
Slide 4
The Right Hand Sides: Shadow Price (Dual Price)
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A shadow price for a right hand side value (or resource
limit) is the amount the objective function will change per
unit increase in the right hand side value of a constraint.
The range of feasibility for a change in the right hand side
value is the range of values for this coefficient in which the
original shadow price remains constant.
Graphically, a shadow price is determined by adding +1 to
the right hand side value in question and then resolving
for the optimal solution in terms of the same two binding
constraints.
The shadow price is equal to the difference in the values of
the objective functions between the new and original
problems.
The shadow price for a non-binding constraint is 0.
Slide 5
Example 1
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Refer to the “Woodworking” example of Chapter 2,
where X1 = Tables and X2= Chairs. The problem is
shown below.
Max. Z=
s.t.
$100X1+60X2
12X1+4X2 < 60 (Assembly time in hours)
4X1+8X2 < 40 (Painting time in hours)
The optimum solution was X1=4, X2=3, and Z=$580.
Answer the following questions regarding this
problem.
Slide 6
Answer the following Questions:
1. Compute the range of optimality for the contribution of X1 (Tables)
2. Compute the range of optimality for the contribution of X2 (Chairs)
3. Determine the dual Price (Shadow Price) for the assembly stage.
4. Determine the dual Price (Shadow Price) for the painting stage.
Slide 7
Standard Computer Output
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Software packages such as Microsoft Excel and LINDO
provide the following LP information:
Information about the objective function:
its optimal value
coefficient ranges (ranges of optimality)
Information about the decision variables:
their optimal values
their reduced costs
Information about the constraints:
the amount of slack or surplus
the dual prices
right-hand side ranges (ranges of feasibility)
Slide 8
Sensitivity Report
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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.
Slide 9
Example 2: Olympic Bike Co.
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Model Formulation
Max
s.t.
10x1 + 15x2
(Total Weekly Profit)
2x1 + 4x2 < 100 (Aluminum Available)
3x1 + 2x2 < 80 (Steel Available)
x1, x2 > 0
(Non-negativity)
Slide 10
Example 2: Olympic Bike Co.
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Optimal Solution
According to the output: x1 (Deluxe frames) = 15,
x2 (Professional frames) = 17.5, and the objective
function value = $412.50.
Slide 11
Example 2: Olympic Bike Co.
Range of Optimality
Question
Suppose the profit on deluxe frames is increased to $20.
Is the above solution still optimal? What is the value of
the objective function when this unit profit is increased
to $20?
Answer
The output states that the solution remains optimal as
long as the objective function coefficient of x1 is between
7.5 and 22.5. Since 20 is within this range, the optimal
solution will not change. The optimal profit will
change: 20x1 + 15x2 = 20(15) + 15(17.5) = $562.50
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Slide 12
Example 2: Olympic Bike Co.
Range of Optimality
Question
If the unit profit on deluxe frames were $6 instead of
$10 would the optimal solution change?
Answer
The output states that the solution remains optimal as
long as the objective function coefficient of x1 is
between 7.5 and 22.5. Since 6 is outside this range, the
optimal solution would change.
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Slide 13
Example 2: Olympic Bike Co.
Range of Feasibility: The range of feasibility for a change
in a right-hand side value is the range of values for this
parameter in which the original shadow price remains
constant.
Question
What is the maximum amount the company should pay
for 50 extra pounds of aluminum?
Answer
The shadow price provides the value of extra aluminum.
The shadow price for aluminum is $3.125 per pound and
the maximum allowable increase is 60 pounds. Since 50 is
in this range, then the $3.125 is valid. Thus, the value of
50 additional pounds is: 50($3.125) = $156.25
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Slide 14
Example 3
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Consider the following linear program:
Min 6x1 +
s.t.
9x2
x1 + 2x2
10x1 + 7.5x2
x2
x1, x2
($ cost)
<
>
>
>
8
30
2
0
Solve the above problem and perform sensitivity
analysis.
Slide 15
Range of Optimality and 100% Rule
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The 100% rule states that simultaneous changes in
objective function coefficients will not change the
optimal solution as long as the sum of the percentages
of the change divided by the corresponding maximum
allowable change in the range of optimality for each
coefficient does not exceed 100%.
Slide 16
Range of Feasibility
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A dual price represents the improvement (increase or
decrease) in the objective function value per unit
increase in the right-hand side. As long as the righthand side remain within the range of feasibility, there
will be is no change in the shadow price. The range of
feasibility is the range over which the dual price is
applicable.
Slide 17
Range of Feasibility and 100% Rule
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The 100% rule states that simultaneous changes in
right-hand sides will not change the dual prices as long
as the sum of the percentages of the changes divided by
the corresponding maximum allowable change in the
range of feasibility for each right-hand side does not
exceed 100%.
Slide 18
Reduced Cost
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The reduced cost for a decision variable whose value
is 0 in the optimal solution is the amount the
variable's objective function coefficient would have to
improve (increase for maximization problems,
decrease for minimization problems) before this
variable could assume a positive value.
The reduced cost for a decision variable with a
positive value is 0.
Slide 19
Example 4
In a product-mix-problem, X1, X2, X3, and X4 indicate the
units of products 1, 2, 3, and 4, respectively, and the linear
programming model is
MAX Z = $5X1+$7X2+$8X3+$6X4
S.T.
1) 3X1+2X2+4X3+3X4600
2) 4X1+1X2+2X3+6X4700
3) 2X1+3X2+1X3+2X4800
Machine A hours
Machine B hours
Machine C hours
Input the data in an Excel file and save the file. Using
Solver, solve the problem and obtain sensitivity results.
Slide 20
Summary
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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.
Slide 21