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REVISED SIMPLEX METHODRevised Simplex Method (5.2,5.3,5.4)

Dual Introduction (6.1)

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Revised Simplex Method

• Method was motivated for efficient computational purposes

• Works only with relevant piece of information at each iteration

• Revised Simplex Method uses Matrix Notation and Matrix

Manipulations• The insights it provide us are very useful in dual formulation of

the original LP (which is called primal)

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Matrix Notation

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Matrix Notation : Augmented Form

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Matrix Notation : Augmented Form

Example – WYNDOR Glass Co.

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Matrix Notation : Current set of equations

(Simplex Tableau)

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Current Set of Equations


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Solving for BFS with Revised Simplex

• Eliminate non basic variables from

to obtain the vector of basic variables

• Eliminate a j columns of Non Basic Variables from

to obtain the basis matrix

We want to work only with Basic Variables

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Solving for BFS with Revised Simplex

• Basic Variable Values are obtained by:

• C B is the vector whose elements are objective function

coefficients corresponding to X B then the value of objectivefunction for the current basic solution is

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Example - WYNDOR

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Getting all the other values of current iteration

through B-1

•

Matrix form of current set of equations:

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Premultiply both sides with


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Simplex tableau in Matrix Form

• At any iteration:

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Simplex Tableau – Fundamental Insights

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• New row(1) = old row (1)

• New row(2) = ½ *old row( 2)

• New row (3) = (-1)* old row (2) + 1*old row (3)

• After any iteration , the coeff. of the slack variables in each equation immediately reveal how that equation hasbeen obtained from the initial equations.

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Current itr. Slack Coeff Initial rows 1-3

=

New rows 1-3


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=

New row 0

+

Initial row 0 initial rows 1 to 3Currentslack vari.

cost coeff


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Revised Simplex Algorithm

• Initialize the Problem, Introduce slack variables

• Obtain

• Iteration

• Determine the Entering Basic Variable

• Determine the Leaving Basic Variable

• Determine the new BFS

• Optimality Test

• Current solution is optimal if all the coeff are non negative

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Revised Simplex – Summary

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Dual - Introduction

• At any iteration simplex method for primal problem, current

Row 0 is represented as shown in the table

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Dual - Introduction

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y

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Dual - Introduction

Iterations Primal Dual

0 Feasible, Not optimal Infeasible

1 Feasible, Not optimal Infeasible

2 Feasible, Optimal Feasible, Optimal

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PRIMAL - DUAL

Dual Formulation (6.1)Primal-Dual Relations (6.1, 6.3)

Dual Solution (Dual Simplex – 7.1)

Sensitivity Analysis (6.5)

Economic Interpretations (6.2)

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Dual - Introduction

• Every Linear Programming problem is associated with it

another LP called the Dual , the original is called the Primal

• When ever we solve LP we are solving two problems

• Primal Resource Allocation Problem

• Dual Resource Valuation Problem

• Primary use of Duality lies in the interpretation and

implementation of sensitivity analysis – How to trade off

resources

• If Primal has n variables and m constraints, its Dual has m

variables and n constraints

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Dual Formulation

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Dual Formulation

• Number of parameters remain the same

• Primal obj. coeff. become constraint right hand sides in Dual

• Primal constraint RHS becomes obj. coeff. in Dual

• Coeff. of variable in functional constraint of primal problem are

the coeff in a functional constraint of the Dual

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Dual Formulation: Rules *

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* Taken from Hamdy A. Taha

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Dual Formulation: Example 1

• Try to formulate dual of the dual*

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Dual Formulation: Example 2

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Primal – Dual variables correspondence

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Primal – Dual Relationships

Theorem Primal Dual

Weak Duality Property: If x is a

feasible solution for the primal

problem and y is a feasible solution

for the dual problem, then

cx ≤ yb

x1 = 3

x2 = 3

Z = cx = 24

y1 = 1

y2 = 1

y3 = 2

W = yb = 36

Strong Duality Property: if x* is an

optimal solution for the primal

problem and y* is an optimal solution

for the dual problem, then

cx* = y*b

x1 = 2

x2 = 6

Z = cx *= 36

(Primal Maximumfeasible hence optimal)

y1 = 0

y2 = 3/2

y3 = 1

W = y*b = 36

(Dual Minimum

feasible hence

optimal)

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Primal – Dual Relationships

Theorem Primal Dual

Complementary Solutions Property: At each

iteration , the simplex method simultaneously

identifies a CPF solution x for primal and a

complementary solution y for its dual

problemcx = yb

x1 = 0

x2 = 6

Z = cx = 30

y1 = 0

y2 = 5/2

y3 = 0

W = yb = 30

Complementary Optimal Solutions Property:

At the final iteration, the simplex method

simultaneously identifies an optimal solution

x* for the primal problem and a

complementary optimal solution y* for the

dual problem

cx* = y*b

x1* = 2

x2* = 6

Z *= cx* = 36

y1* = 0

y2* = 3/2

y3* = 1

W* = y*b = 36

/ / 30 p g

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Primal – Dual Relationships (Basic Solutions)

Theorem Primal Dual

Complementary Basic Solutions Property:

Each basic solution in the primal problem has a

complementary basic solution in the dual

problem, where their respective objectivefunction values (Z and W) are equal.

(Basic)

x3 = 4

x4 = 12

x5 = 18

(Non basic)x1 = 0

x2 = 0

(Non basic)

y1 = 0

y2 = 0

y3 = 0

(Basic)y4 = -3

y5 = -5

Complementary Slackness Property:

The primal basic solution and the complementary

dual basic solution satisfy the complementary

slackness relation

i.e, Basic variables in primal are Non basic

variables in dual and Vice versa

Primal Basic

(0,6,4,0,6)

Basic: x2,x3,x5

Non Basic

x1,x4

Compl. Dual

basic

(0,5/2,0,-3,0)

Non Basic

y1,y3,y5 = 0

Basic

(y1 + 3y3 - 3)

(2y2 + 2y3 - 5)

/ / p g

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Primal - Dual Relationships

/ / p g

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Simplex on the Dual

Y1 Y2 Y3 Y4 a1 Y5 a2 Z Coeff Mi

Z -1 -9996 12 -29982 10000 0 0 10000 -30000 0

a1 0 1 0 3 -1 1 0 0 3

a2 0 0 2 2 0 0 -1 1 5


Z -1 -9996 -19988 -49982 10000 0 10000 0 -80000 -49982

a1 0 1 0 3 -1 1 0 0 3 1

a2 0 0 2 2 0 0 -1 1 5 2.5


Z -1 6664.667 -19988 0 -6660.666667 16660.67 10000 0 -30018 -19988

Y3 0 0.333333 0 1 -0.333333333 0.333333 0 0 1Y2 0 -0.66667 2 0 0.666666667 -0.66667 -1 1 3


Z -1 2 0 0 2 9998 6 9994 -36 0

Y3 0 0.333333 0 1 -0.333333333 0.333333 0 0 1

Y2 0 -0.33333 1 0 0.333333333 -0.33333 -0.5 0.5 1.5

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Dual Simplex

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Why Dual?

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TO BE UPDATED…

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4 - Primal and Dual V1