The Theory of the Simplex Method

Foundations-1

The Theory of the Simplex MethodThe Theory of the Simplex Method

Foundations-2

The EssenceThe Essence

• Simplex method is an algebraic procedure• However, its underlying concepts are geometric• Understanding these geometric concepts helps before

going into their algebraic equivalents• With two decision variables, the geometric concepts are

easy to visualize• Should understand the generalization of these concepts

to higher dimensions

Foundations-3

Standard (Canonical) Form of an LP ModelStandard (Canonical) Form of an LP Model

Maximize Z = c1x1 + c2x2 + … + cnxn

subject to a11x1 + a12x2 + … + a1nxn ≤ b1

a21x1 + a22x2 + … + a2nxn ≤ b2

…am1x1 + am2x2 + … + amnxn ≤ bm

x1 ≥ 0, x2 ≥ 0, …, xn ≥ 0

Foundations-4

Extended TerminologyExtended Terminology

max cx n variabless.to Ax b m inequalities

x 0 n inequalities

max cx n+m variabless.to Ax + Ixs = b m equations

x, xs 0 n+m inequalities

• Constraint boundary equation– For any constraint, obtain by replacing its , =, by =– It defines a “flat” geometric shape: hyperplane

• Corner-point solution– Simultaneous solution of n constraint boundary equations

( )

Foundations-5

Extended TerminologyExtended Terminology

max cx + 0xs n+m variabless.to Ax + Ixs = b m equations

x, xs 0 n+m inequalities

• Indicating variables (in the augmented form)

Original Constraint in Augmented Form

Constraint Boundary Equation Indicating Variable

xj 0 (j=1,2,…,n)

xj = 0 xj

(i=1,2,…,m)

xsn+iisin

n

jjij bxxa

1i

n

jjij bxa

1

Foundations-6

' '1 2 3 3

1 11

2 22

3 ' ' 33

Max 0 0 variables

s.to = equations

= equations

- + equations

, ,

s a s a

s

a

s a

s a

cx x Mx x Mx n m m m m

A x Ix b m

A x Ix b m

A x Ix Ix b m

x x x ' '1 2 3 3, , 0 inequalities s ax x n m m m m

Foundations-7

Original Constraint

Original Constraint in Augmented Form

Constraint Boundary Equation

Indicating Variable

xj 0 xj 0 xj = 0 xj

(i=1,2,…,m1)

(i=1,2,…,m2)

(i=1,2,…,m3)

Indicating variables, overallIndicating variables, overall

' ' s an i n ix x

sn ix1 1

1

n

sij j n i i

j

a x x b 1 1

1

n

ij j ij

a x b1 1

1

n

ij j ij

a x b

2 2

1

n

ij j ij

a x b

3 3

1

n

ij j ij

a x b

2 2

1

n

aij j n i i

j

a x x b

3 ' ' 3

1

n

s aij j n i n i i

j

a x x x b

2 2

1

n

ij j ij

a x b

3 3

1

n

ij j ij

a x b

an ix

Foundations-8

Properties of CPF (BF) SolutionsProperties of CPF (BF) Solutions

Property 1:a. If there is exactly one optimal solution, then it must be a

corner-point feasible solutionb. If there are multiple optimal solutions (and a bounded

feasible region), then at least two must be adjacent corner-point feasible solutions

Proof of (a) by contradiction

Foundations-9


Property 2:There are only a finite number of corner-point feasible solutions


x 0 n inequalities

CP solution is defined as the intersection of n constraint boundary equations(n+m) choose n:

!m!n)!mn(

nmn

Upper bound on # of CPF solutions

Foundations-10


Property 3:a. If a corner-point feasible solution has no adjacent CPF

solution that are better, then there are no better CPF solutions elsewhere

b. Hence, such a CPF solution is an optimal solution (assuming LP is feasible and bounded)

Foundations-11

Simplex in Matrix (Product) Form Simplex in Matrix (Product) Form


x 0 n inequalities

max cx m+n variabless.to Ax + Ixs = b m equations

x, xs 0 m+n inequalities

Initial Tableau:

( )B.V.

Original Variables Slack Variablesr.h.s.

x1 x2 … xn xsn+1 … xs

n+m

Z -c 0 0xs

n+1

A I b…

xsn+m

XB=

Foundations-12

Simplex in Matrix (Product) Form Simplex in Matrix (Product) Form

Intermediate iterations:• Let B denote the square matrix that contains the columns from

[ A | I ] that correspond to the current set of basic variables, xB (in order)

• Similarly, let cB be the vector of elements in c that correspond to xB

• Then at any intermediate step, the simplex tableau is given by

B.V.Original Variables Slack Variables

r.h.s.x1 x2 … xn xs

n+1 … xsn+m

Z cBB-1A-c cBB-1 cBB-1b

xB B-1A B-1 B-1b

Foundations-13

Revised Simplex MethodRevised Simplex Method

• InitializationFind an initial BFS (if one not immediately available, do Phase 1. If Phase 1 terminates with z1>0, LP is infeasible)

• Optimality testCalculate cBB-1A-c for basic, and cBB-1 for nonbasic variables. If all ≥ 0, stop with optimality

• Iterative step– Determine the entering variable as before (steepest ascent)– Let xk be the entering variable– Determine the leaving variable as before (minimum ratio test), but now only need to

calculate the coefficients of the pivot column (from B-1A or B-1) and the updated rhs (B-1b)– Let xl : the leaving variable

r : the equation number of the leaving variablea’rk : coefficient of xk in rth equation (the pivot element)

– Determine B-1new=E B-1

old where E is I expect for rth column replaced with

– Can calculate the new BFS using xB=B-1b and z=cBB-1b – Back to optimality test

ri if'a1

ri if'a'a

,

rk

rk

ik

i

m

1

Foundations-14

Revised Simplex ExampleRevised Simplex Example

Original data:

Maximize z = 2x1+ 3x2

subject to x1 + 2x2 +s1 = 103x1+ x2 +s2 = 15

x2 +s3 = 4

x1,x2, s1, s2, s3 ≥ 0

41510

b 101321

A

32c

Maximize z = 2x1+ 3x2

subject to x1 + 2x2 ≤ 10

3x1+ x2 ≤ 15

x2 ≤ 4

x1,x2 ≥0

Foundations-15

Iteration 0Iteration 0

bBcz

bBx

B B x

1-B

1-B

1-B

Foundations-16


• Calculate row-0 coefficients:cBB-1A-c =

cBB-1 =

Optimal?• Entering variable is• Calculate coefficients in pivot column using B-1A or B-1

• Calculate updated rhs using B-1b, do the ratio test

• Leaving variable is• Update B-1

Foundations-17


bBcz

bBx

B B x

1-B

1-B

1-B

Foundations-18



cBB-1 =




Foundations-19


bBcz

bBx

B B x

1-B

1-B

1-B

Foundations-20



cBB-1 =




Foundations-21


bBcz

bBx

B B x

1-B

1-B

1-B

Foundations-22



cBB-1 =




The Theory of the Simplex Method

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