linearprogrammingppt-130902003046-phpapp02

8/12/2019 linearprogrammingppt-130902003046-phpapp02

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LINEAR

PROGRAMMING

Presented By Meenakshi Tripathi


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Linear Programming Linear programming (LP , or linear optimization )

is a problem ofmaximizing or minimizing a linear function (objective function) in presence of linear

inequality and/or equality constraints . Standard & Canonical Form

. ; , 0

. ; , 0

Objective function & Constraints Corner Point : x=(x 1 , x 2 ,,x n ) is a vertex iff columns of Ai are Linearly Independent

rank(A)=n and x i 0. Also called Basic feasible solution

Surplus/Slack Variable: Used to transform and inequality into equality eg.

1) = as += + 0

2) = as += + 0


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BASIC and BASIC FEASIBLE SOLUTION

System & 0 , , . If rank(A)=rank(A,b)=m then , , . _1 & 0 .

Then x " " .

If xB0 then x is Basic feasible solution , xB are basic variables and x N nonbasic matrix.

Extreme Points : A point in a convex set is called an extreme point of X, if x cannot berepresented as a strict convex combination of two distinct points in X. i.e. If 11 2 0,1 & 1, 2 , 1 2

Bounded Set: A set is bounded if . . <

Half-spaces : Polyhedral Set/Polyhedron: Intersection of finite-number

of half-spaces. A bounded polyhedral set is called Polytope

Polytope: ,

x1

x2

x3


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Geometric Solution

Interchanging Maximization & Minimization Problems: = =

Types of Geometric Solution:

1) Unique Optimal Solution- unique optimal solution2) Alternative Optimal Solutions- optimal solution set is unbounded3) Unbounded Optimal Objective Value both feasible region & optimal valueunbounded, hence no optimal solution exists4) Empty Feasible Region inconsistent system of equations


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Definitions For positive variables since they represent physical quantities, if unrestricted sign

variable j, replace it with x j x j, where x j 0 and x j 0.

Work from vertex to vertex of the polyhedron and in each step improve the objectivefunction value.

Degeneracy: A LP max* : , } is degenerate if s.t. more thann constraints of that are active at x*

Optimal Bases: Basis B is optimal if it is feasible & unique with TA & i=0, i B . Then x*=A -1BbB is optimal solution of LP.

Adjacent Vertices: Vertices x 1 & x2 of P * : } are adjacent if (n-1)Linearly Independent inequalities active at both x 1 & x2

x* - 3 constraints active


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SIMPLEX ALGORITHMBasic Idea:Start with vertex x*While x* is not optimal

Find vertex x adjacent to x* with cTx > *

update x*:=x

or assert that LP is unbounded


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SIMPLEX ALGORITHM: Basis notation

Basic Idea:

Start with feasible basis BWhile B is not optimal

Let i B be index with i 0

if K= assert LP is unbounded

else

Let k K index whereb k

is attained

Update B:=B\{i} {k}


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SIMPLEX Standard Maximization form of Objective function, constraints in form of less than or

equal to equations, RHS values always positive

Table constructed with basic variables, coefficients of variables, RHS constants & ratiocolumns

0 ( 0)

New Pivot =(old/leaving new element)/key element

Ratio= constant/entry column coefficient

Old element (OE)=OE-(starting column coefficient corresponding new pivot element)


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Example Z=30x 1+20x 2s.t. 2x 1+x2+s 1=100 X1+x2+s 2=80 X1+s 3=40 X1,x2,s 1,s 2,s 30

Iteration 1: column = minimum negative entry = -30; Row : min{ 100/2, 80/1,40/1}=40 => s3 leaving, x1 entering

BV z x1 x2 s1 s2 s3 B

Z 1 -30 -20 0 0 0 0

S1 0 2 1 1 0 0 100S2 0 1 1 0 1 0 80

S3 0 1 0 0 0 1 40

BV z x1 x2 s1 s2 s3 BZ 1 0 -20 0 0 30 1200

S1 0 0 1 1 0 -2 20

S2 0 0 1 0 1 -1 40

x1 0 1 0 0 0 1 40


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Example


Z 1 0 0 20 0 -10 1600

x2 0 0 1 1 0 -2 20

S2 0 0 0 -1 1 1 20

x1 0 1 0 0 0 1 40

Iteration 2: column = minimum negative entry = -20; Row : min{ 20/1, 40/1,-}=20 => s1 leaving, x2 entering

Iteration 3: column = minimum negative entry = -10; Row : min{-, 20/1, 40/1,-}=20 => s2 leaving, s3 entering


Z 1 0 0 10 10 0 1800

x2 0 0 1 -1 2 0 60

S3 0 0 0 -1 1 1 20

x1 0 1 0 0 -1 1 20

All nonbasic variables with non-negative coefficients in row 0,

Optimal solution : x1=20, x 2=60 & Z=1800


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PRIMAL-DUAL

With every LP associated LP called dual

. ; , 0Dual : . ; , 0

orPrimal : . ; , 0 Dual: . ; , 0

Theorems: Dual of dual is primal

Weak Duality: LP max* : , } & its dual min* : y , Ty c, y 0 }. Ifx* & y* are primal & dual feasible respectively, then c Tx* bTy*

Strong Duality : LP max* : , } & its dual min* : y , Ty c, y 0 }.If primal is feasible and bounded then primal feasible x* & dual feasible y* respectively,then c Tx* =b Ty*


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Primal Dual Problem 1: Maximize ZZ=3x 1+4x 24x 1+2x 2 80

3x 1+5x 2 180

Dual of Problem1 : Minimize ZZ=80y 1+180y 24y 1+3y 23 2y 1+5y 2 4


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Reference Linear Programming and network flows, M.S. Bazaraa,

J.J.Jarvis & H.D.Sherali Coursera lectures,Linear Optimization

Wikipedia

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