This document discusses the duality of a linear programming problem and obtaining the dual problem. It provides an example primal problem with an unrestricted variable x3. To solve this, x3 is written as x31 - x32, making it restricted. The dual problem is then derived by associating dual variables with each primal constraint and writing the dual objective function and constraints. The dual problem is provided, along with combining two dual variables y3 and y4 into a single unrestricted variable y. In summary, the document provides an example of obtaining the dual of a primal LP problem with an unrestricted variable.

1. MB106
QUANTITATIVE TECHNIQUES
MODULE I
LECTURE 9
Duality of LP and its interpretation continued
PROF. KRISHNA ROY

2. lpp-Unrestricted variables in duality
Example:
Obtain the dual of the following primal problem
Minimize Z=3x1 -2x2 +x3
Subject to the constraints
2x1-3x2+x3 ≤ 5 (1)
4x1-2x2≥ 9 (2)
-8 x1+4x2 +3 x3 = 8 (3)
x1 ≥ 0, x2 ≥ 0, x3 is unrestricted
As x3 is unrestricted let x3 = x31 - x32 where x31 ≥ 0, x32 ≥ 0
09-11-2021 Prof. Krishna Roy, Dr. B. C. Roy Engineering College 2

3. lpp-Unrestricted variables in duality
On substituting x31 - x32 for x3 and considering equality constraints
the given problem becomes
Minimize Z=3x1 -2x2 + x31 - x32
Subject to the constraints
2x1-3x2+ x31 - x32 ≤ 5 (1)
4x1-2x2≥ 9 (2)
-8 x1+4x2 +3x31 - 3x32 ≥ 8 (3)
-8 x1+4x2 +3x31 - 3x32 ≤ 8 (4)
x1 ≥ 0, x2 ≥ 0, x31 ≥ 0, x32 ≥ 0
09-11-2021 Prof. Krishna Roy, Dr. B. C. Roy Engineering College 3

4. lpp-Unrestricted variables in duality
Multiplying constraints 1 and 4 by -1 to convert ≤ to ≥ we get
Minimize Z=3x1 -2x2 + x31 - x32
Subject to the constraints
-2x1+3x2- x31 + x32 ≥ -5 (1)
4x1-2x2≥ 9 (2)
-8 x1+4x2 +3x31 - 3x32 ≥ 8 (3)
8 x1-4x2 -3x31 + 3x32 ≥ -8 (4)
x1 ≥ 0, x2 ≥ 0, x31 ≥ 0, x32 ≥ 0
09-11-2021 Prof. Krishna Roy, Dr. B. C. Roy Engineering College 4

5. lpp-Unrestricted variables in duality
Hence the dual of the problem is
Maximize W=-5y1 + 9y2 +8y3 -8y4
Subject to the constraints
-2y1 + 4y2 -8y3 +8y4 +≤ 3 (1)
3y1 -2y2 +4y3 -4y4 ≤ -2 (2)
-y1 +3y3 -3y4 ≤ 1 (3)
y1 -3y3 +3y4 ≤ -1 (4)
y1 ≥ 0, y2 ≥ 0, y3 ≥ 0 , y4 ≥ 0 , y5 ≥ 0  non negativity
restrictions
where y1, y2, y3, y4 are dual variables associated with primal
constraints 1,2,3 and 4 respectively
09-11-2021 Prof. Krishna Roy, Dr. B. C. Roy Engineering College 5

6. lpp-Unrestricted variables in duality
Putting y3 - y4 = y we get
Maximize W=-5y1 + 9y2 +8y
Subject to the constraints
-2y1 + 4y2 -8y ≤ 3 (1)
3y1 -2y2 +4y ≤ -2 (2)
-y1 +3y ≤ 1 (3) - - -y1 +3y = 1
y1 -3y ≤ -1 (4)
y1 ≥ 0, y2 ≥ 0, y is unrestricted
where y1, y2, yare dual variables associated with primal constraints
1,2 and 3 respectively
09-11-2021 Prof. Krishna Roy, Dr. B. C. Roy Engineering College 6

7. • Till we meet again in the next class……….
PROF. KRISHNA ROY, FMS, BCREC 7
09-11-2021