BU.520.601
Business Analytics
• Sensitivity Analysis:
• Duality and Sensitivity Analysis
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PRIMAL AND DUAL
PRIMAL
DUAL
Max
Min
Variables:
X1, X2, X3,….
Standard Constraints: ≤
Variables:
Y1, Y2, Y3,…
Standard Constraints: ≥
Objective Functions’
Coefficients
Constraints’ RHS (Known
Number)
RHS’ Known Numbers
Objective Functions’
Coefficient
Optimum Solution Z*
achieved by X1*, X*2, X*3,…
Optimum Solution Z* is
achieved by, Y1*, Y2*, Y3*,..
Linear Optimization
LP: Sensitivity Analysis
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EXMPLE 1: PRIMAL AND DUAL
Linear Optimization
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EXAMPLE 2: Primal Dual Linear Programs
and Their Relationship
PRIMAL LP
DUAL LP
Max Z= 7X1 + 11 X2
St.
1X1 + 5 X2  10
Min Z = 10Y1+6Y2+8Y3
St.
4X1 + 3 X2  6
1 Y1 +4 Y2 +2 Y3 ≥ 7
5 Y1 +3 Y2+ 90 Y3 ≥ 11
2X1+ 90 X2  8
X1, X2 ≥ 0
Optimum Solution Z* is achieved by
X*1, X*2, X*3 ….
Y1, Y2, Y3 ≥ 0
Optimum Solution Z* is achieved by
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Y*1, Y2*, Y3*,...
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Economists’ Interpretation of dual variables
The dual program is that, in LP, the Decision Variables of the Dual
problem have the following Economic interpretation:
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Application of Duality in Economic:
Dual Price or Shadow Price in Economics:
By finding the dual of a primal linear program, we want to find out:
1. The improvement in the Objective Function’s value (Z) that
results from One-Unit change in its Right Hand Side
(Available Resources). That is, by changing the Right Hand
side of its (Primal ‘s) Constraint.
2. The improvement in the Objective Function’s value (Z) that
results from One-Unit change in unit Cost of its Decision
Variables (Coefficients C1, C2, C3, …..in Z = C1 X1 + C2 X2
+C3 X3 +… ).
Note: We makes changes in the Objective Function’s Coefficients or Changes in
any Constraint’s RHS Coefficients – One coefficient at a time.
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BU.520.601
on Application of Duality in Economic:
Dual Price or Shadow Price in Economics: Shadow Prices
• The Dual Program’s Optimum Solutions Y*1, Y*2,…. Are called
Shadow Prices in Economics.
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Introduction to Sensitivity Analysis
Sensitivity analysis means determining effects of changes in
a parameter ( here, C1, C2, C3, in the Objective Function or changes
in the Right Side of any Constraint) on the solution.
Note: Sensitivity Analysis, is also called What if Analysis, Parametric
Analysis, or Post Optimality Analysis. It is not restricted to LP
problems.
Consider EXAMPLE 3 again on Next Page,
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EXAMPLE 3: Applications of Dual Programs in Sensitivity Analysis.
A manufacturer makes two types of golf bags: Deluxe & Ace.
Profit/unit: $10 (for D) and $8 (for A). Resources allocations, constraints, and
consumptions numbers are given in following Table:
Operation
Cut & Dye
Sew
Finish
Inspect & pack
Deluxe (D)
Hrs./ unit
7/10
1/2
1
0.1
Ace (A)
Hrs. / unit Max. Hrs. / week
1
630
5/6
600
2/3
708
1/4
135
Weekly total weekly production should be at least 150 units.
The Question is, how many units of “Deluxe (D)” and Ace (A)” should be
produced per week to maximize profit? Assume all units can be sold.
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EXAMPLE 3 (Continued )
Consider the LP problem shown. We will call this a
primal problem. Find its Dual problem.
Max Z = 10x1 +
8x2
x2 ≤
630
(5/6) x2 ≤
600
(2/3) x2 ≤
708
(1/4) x2 ≤
135
x2 ≤
-150
x1 ≥ 0, x2 ≥ 0
0.7x1 +
(½) x1 +
x1 +
(1/10) x1 +
-x1 -
Solution: Its Dual will be the following Minimization problem:
Min Z = 630y1 + 600y2
+
0.7y1 + (½)y
708y3 +
135y4
-
150y5
+ y3 +
(1/10)y4
-y5 ≥
10
(2/3)y3 +
(1/4)y4
-y5 ≥
8
2
y1 + (5/6)
y2
+
y1 ≥ 0, y2 ≥ 0, y3 ≥ 0, y4 ≥ 0, y5 ≥ 0
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Note the following in Example 3
•Any one of these two problems can be called “primal”; the other one is
“dual.”
•If we solve one, we implicitly solve the other.
•The optimum values of both Primal and Dual problems are equal (Z*)
Primal Max Z* = 7416
optimal
Dual Min Z* = 7416
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Sensitivity Analysis
Objective function
Maximize Z= 10 x1 + 8 x2
 7/10 x1 +
x2  630
 1/2 x1 + 5/6 x2  600
 x1 + 2/3 x2  708
 1/10 x1 + 1/4 x2  135
 x1 +
x2 ≥ 150
x1 ≥ 0, x2 ≥ 0
We will consider changes in the objective
function’s Coefficients or the RHS coefficients –
one coefficient at a time. Then, we try to
determine the corresponding change in the value
of Objective Function (Z).
Right Hand Side (RHS).
Optimal solution:
x1 = 540, x2= 252.
Z = 7416
Here are some questions we will try to answer.
Q1 What happens if unit profit of Ace can go up from $8 per unit of Decision
Variable X2 to $9 or it goes down from $ 8 per unit cost to $7 per unit?
Q2:What if per unit profit for Deluxe model (Decision Variable X1 ) changes
from $10 to $ 11?
Q3: What if 10 more hours of production time is available in Constraints 
cutting & dyeing?  inspection? That
is, if we Change: 630 to 640 & 135 to
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145? (Do it by changing one number at a time)
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