Internet Engineering
Czesław Smutnicki
Discrete Mathematics – Linear Programming
CONTENTS
• Linear programing
• Simplex method
FORMULATION
max x x0  cx
Ax  b
x0
c1n , xn1, Amn , bm1
PROPERTIES
A  ( B, N ), Bmm , Nm(nm)
x  ( xB , xN ), c  (cB , cN )
Bx B  Nx N  b
xB  B 1b  B 1 Nx N
x0  cB xB  cN xN
x0  cB B 1b  (cB B 1 N  cN ) xN
 x0  cB B 1b cB B 1N  cN 
 xN
 x    1   
1
B N

 B   B b  
ANALYSIS
B1, B2 ,...,Bm , R, (a j , j  R), xB  ( xB1 , xB2 ,...,xBm ), xB0  x0
 y00 
cB B 1b   y10 
y0   1  
 B b   ... 


 ym 0 
 y0 j 
cB B 1a j  c j   y1 j 
yj  
 
1
 B a j
  ... 
 
 ymj 
xBi  yi 0   jR yij x j , i  0,1,2,...,m
x0  xB0  y00   jR y0 j x j
y0 j  0, j  R  y00  max, ( x j  0, j  R)
k  R, y0k  0  x0  iff xk 
xBi  yi0   jR\{k} yij x j yik xk , i  1,2,...,m
ANALYSIS cont.
xBi  iff xk , yik  0, xk 
yi 0
 vik
yik
vrk  min{ vik : yik  0, i  1,2,..., m}
xk  vrk
xBi  yi 0  yik vrk , i  1,2,..., m
xBr  0
SIMPLEX METHOD
xBr  yr 0   jR \{k } yrj x j  yrk xk
xk 
yrj
yr 0
1
  jR \{k }
xj 
xBr
yrk
yrk
yrk
yrj 

yr 0
y
 x j  ik xB
xBi  yi 0  yik
  jR \{k }  yij  yik
yrk
yrk 
yrk r

Thank you for your attention
DISCRETE MATHEMATICS
Czesław Smutnicki