Illustration on Graphical Method of Linear Programming. Conceptual understanding of Feasibility, Optimal Solution & Convex Sets.

1. Linear Programming
Terminology

2. What is a Mathematical Model ?
F=ma
‘Mathematical Expressions’
o Here m and a are called as ‘Decision Variables’
o F can be called as ‘Objective Functions’
o Now, F can be controlled or restricted by limiting m or
a … say m < 50 kg …here, m can be called as a
‘Constraint’
o Similarly if a > o …always, then this condition is called
as ‘Non-Negativity Condition’
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3. Illustration:
Maximize: Z = 3x1 + 5x2 Objective
Function
Subject to restrictions:
x1 <4
Functional
2x2 < 12 Constraints
3x1 + 2x2 < 18
Non negativity condition
x1 >0 Non-negativity
x2 >0 constraints
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4. What is Linear Programming (LP)?
 The most common application of LP is allocating
limited resources among competing activities in a
best possible way i.e. the optimal way.
 The adjective linear means that all the mathematical
functions in this model are required to be linear
functions.
 The word programming does not refer to computer
programming; rather, essentially a synonym for
planning.
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5. Graphical Solution
Ex) Maximize: Z = 3x1 + 5x2
Subject to restrictions:
x1 < 4
2x2 < 12 i.e. x2 < 6
3x1 + 2x2 < 18
Non negativity condition
x1, x2 > 0
Solution: finding coordinates for the constraints (assuming perfect equality), by putting
one decision variable equal to zero at a time.
Restrictions (Constraints) Co-ordinates
x1 < 4 (4 , 0)
x2 < 6 (0 , 6)
3x1 + 2x2 < 18 (0 , 9) & (6 , 0)
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6. Restrictions (Constraints) Co-ordinates Non-negativity Constraint
x1 < 4 (4 , 0) x1, > 0
x2 < 6 (0 , 6) x2 > 0
3x1 + 2x2 < 18 (0 , 9) & (6 , 0)
X2
10
8
A B
6
4
C Feasible Region (Shaded / Points A, B, C, D and E)
2
0 D
E 2 4 6 8 10 X1
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7. Feasible Solutions
 Try co-ordinates of all the corner points of
the feasible region.
 The point which will lead to most
satisfactory objective function will give
Optimal Solution.
 Note: for co-ordinates at intersection; solve
the equations (constraints) of the two lines
simultaneously.
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8. Optimal Solution
Corner Limiting Constraint Co-ordinate Max. Z= 3x1 + 5x2
A x2 = 6 (0 , 6) 30
B x2 = 6 & 3x1 + 2x2 = 18 (2 , 6) 36
C x1 = 4 & 3x1 + 2x2 = 18 (4 , 3) 27
D x1 = 4 (4 , 0) 12
E Origin (0 , 0) 0
From the above table, Z is maximum at point ‘B’ (2 , 6) i.e. The
Optimal Solution is:
X1 = 2 and
X2 = 6 ANSWER
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9. What is Feasibility ?
 Feasibility Region
[Dictionary meaning of feasibility is possibility]
“The region of acceptable values of the
Decision Variables in relation to the
given Constraints (and the Non-Negativity
Restrictions)”
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10. What is an Optimal Solution ?
 It is the Feasible Solution which Optimizes.
i.e. “provides the most beneficial result for the specified
objective function”.
 Ex: If Objective function is Profit then Optimal Solution
is the co-ordinate giving Maximum Value of „Z‟…
While; if objective function is Cost then the optimum
solution is the coordinate giving Minimum Value of „Z‟.
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11. Convex Sets and LPP’s
“If any two points are selected in the feasibility region
and a line drawn through these points lies completely
within this region, then this represents a Convex Set”.
Convex Set Non-convex Set
A
A
B B
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