Duality Theory Operational Research Level 4

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Duality Theory Operational Research Level 4

• Prepared by T.M.J.A.Cooray
• Department of Mathematics

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Introduction

• Associated with every LP problem there is
  another LP problem called a dual problem
• The original problem is called the Primal
  Problem and the other is called the Dual
  Problem

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• Primal problem
• Min ZXc1X1CnXn
• Subject to
• a11X1a12X2a1nXn ?b1
• .
• ai1X1ai2X2.ainXn ?bi
• ..
• am1X1am2X2amnXn ?bm
• X1,X2, Xn ? 0

• Dual problem
• Max Zy b1Y1..bmYm
• Subject to
• a11Y1..am1Ym ? c1
• a12Y1..am2Ym ? c2
• a1jY1a2jY2..am2Ym ? cj
• .
• a1nY1..amnYm ? cn
• Y1,.,Ym ? 0

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Real world example of a Primal and a Dual
problem

• House wifes problem is to give a balanced diet
  to the family with a minimum cost
• The diet should include all necessary nutrients
  , say N1,N2, Ni ,..Nm at least in minimum
  required amounts b1,b2,,bi,...bm.
• There are n types of food items say,F1,F2,..Fj..
  Fn
• Each food having a unit cost of C1,.Cn
  respectively.
• House wife has to determine, how much to buy
  from each of the food items .say X1,.Xn.

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Fj
amj Nm
a1j N1
a2j N2
aij Ni
According to nutrition table one unit of food
item Fj will include the nutrients as given
above.
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Pill manufacturers problem

• Pill manufacturer intends to compete with fresh
  food and give a balanced diet using pills.
• He is planning to produce pills from each type of
  nutrients , that is p1, p2, .pm .
• One pill contains one unit of nutrient.
• To get bi units of nutrient Ni the mother should
  by bi pills of Pi.
• Price of a pill would be Y1,.Ym.
• Determine the price of each pill in order to max
  the profit.

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Relationships between Primal and the
Dual

• Minimization
• R.h.s values
• Aij
• inequalities ?
• Number of Variables

• Maximization
• Objective function coefficients
• Aji
• Inequalities ?
• Number of constraints
• (slack variables)

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Duality theorem 1

• Max Zx Min Zy for all the feasible
  solutions of the two problems.

At the optimal solution Max Zx Min Zy
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Duality theorem 2( complementary slackness
theorem )

• If the slack variable corresponding to a
  constraint of one problem is positive then the
  corresponding variable of the other problem is
  zero .

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• Max ZX6X18X2
• Subject to
• 5X110X2?60
• 4X14X2?40
• X1,X2?0

• Min Zy60Y140Y2
• Subject to
• 5Y14Y2? 6
• 10Y14Y2? 8
• Y1,Y2? 0

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Solution
Y1 Y2
X1 X2
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