LINEAR PROGRAMMING PROBLEM – DUALITY:
Definition:
With every linear programming problem (LPP) there is another intimately
related LPP called the dual problem.
The original LPP is called the primal problem.
According to the duality theorem, “ for every maximisation or minimisation
problem in linear programming, there is a unique similar problem of
minimisation or maximisation involving the same data which describes the
original problem”
The variables of the dual LPP are called dual variables and have important
economic interpretation, which can be used by a decision maker for planning
their resources.
Formulation of Dual Linear Programming Problem:
Consider the general linear programming problem (Primal Problem)
Maximise Z= 𝑐1 𝑥1 + 𝑐2 𝑥2 + ⋯ + 𝑐𝑛 𝑥𝑛 (Objective function)
Subject to Constraints
𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 ≤ 𝑏1 (Constraints)
𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 ≤ 𝑏2
.
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.
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.
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𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 ≤ 𝑏𝑚
𝑥1 ,𝑥2 , … 𝑥𝑛 ≥ 0 (Non -negativity constraints)
The corresponding Dual LPP is expressed as
Minimisation Z= 𝑏1 𝑦1 + 𝑏2 𝑦2 + ⋯ + 𝑏𝑚 𝑦𝑚 (Objective function
𝑎11 𝑦1 + 𝑎21 𝑦2 + ⋯ + 𝑎𝑚1 𝑦𝑚 ≥ 𝑐1 (Constraints)
𝑎12 𝑦1 + 𝑎22 𝑦2 + ⋯ + 𝑎𝑚2 𝑦𝑚 ≥ 𝑐2
.
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.
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𝑎1𝑛 𝑦1 + 𝑎2𝑛 𝑦2 + ⋯ + 𝑎𝑚𝑛 𝑦𝑛 ≤ 𝑐𝑛
𝑦1 ,𝑦2 , … 𝑦𝑚 (Non -negativity constraints)
A.W.Tucker explained the relationship between primal and its dual through the
following table
Dual Variable
Primal Variables
𝑥1 ≥ 0
𝑥2 ≥ 0
𝑥𝑛 ≥ 0
𝑎11
𝑎12
𝑎1𝑛
𝑎21
𝑎22
𝑎2𝑛
𝑎𝑚1
𝑎𝑚2
𝑎𝑚𝑛
≥
≥
≥
𝑐1
𝑐2
𝑐𝑛
𝑦1 ≥ 0
𝑦2 ≥ 0
𝑦𝑚 ≥ 0
Relation
Constraints
Relation
≤
≤
≤
Constraints
𝑏1
𝑏2
𝑏𝑚
Min Zy
Max Zx
Problem:
Write the dual of the following LP problem:
Maximise Z= 45x1 + 80x2
Subject to constraints
5x1 + 20x2≤ 400
10x1 + 15x2≤ 450
x1 , x2≥ 0
Dual Variable
𝑦1 ≥ 0
𝑦2 ≥ 0
𝑦𝑚 ≥ 0
Relation
Constraints
Primal Variables
𝑥1 ≥ 0
𝑥2 ≥ 0
𝑥𝑛 ≥ 0
5
20
𝑎1𝑛
10
15
𝑎2𝑛
𝑎𝑚1
𝑎𝑚2
𝑎𝑚𝑛
≥
≥
≥
45
80
𝑐𝑛
Minimisation Z= 400𝑦1 + 450𝑦2 (Objective function
5𝑦1 + 10𝑦2 ≥ 45 (Constraints)
20𝑦1 + 15𝑦2 ≥ 80
𝑦1 ,𝑦2 ≥ 0
Relation
≤
≤
≤
Max Zx
Constraints
400
450
𝑏𝑚
Min Zy