1

Simplex Method for Bounded Variables

Linear programming problems with lower and upper bounds

Generalizing simplex algorithm for bounded variables

Reference: Chapter 5 in Bazaraa, Jarvis, and Sherali

2

LP Problems with Bounded Variables

In most practical Problems, variables are bounded from below as well as above:

We can handle the upper bound constraints implicitly and thus reduce the size of the basis matrix substationally.

. .

Minimize cx

s t Ax b

l x u

x u

3

Basic Feasible Solution

Consider the system:

Partition A into [B,N1,N2] such that B is an invertible matrix. We call it a basis structure.

The basic solution corresponding to this basis structure is determined in the following manner:

If for all , we say that the solution is a basic feasible solution.

Ax b

l x u

1 2

1

2

11 2

j j

j j

B N N

x l for all j N

x u for all j N

x B b N l N u

j j jl x u j B

4

Basic Feasible Solution (cont’d)

Consider the solution space given by:

Which is equivalent to:

1 2

1 2

1

2

5

2 4

0 4

1 4

x x

x x

x

x

1 2 3

1 2 4

1

2

3

4

5

2 4

0 4

1 4

0

0

x x x

x x x

x

x

x

x

5

Basic Feasible Solution (cont’d)

Suppose that x2 and x4 are made basic variables, then:

1) Suppose we set x1=0 and x3=0. Then x2=5 and x4=-6. The resulting solution is not a basic feasible solution.

2) Suppose we set x1=4 and x3=0. Then x2=1 and x4=6, which is a basic feasible solution.

In general, what is the maximum number of basic feasible solutions we can have?

2 1 3

4 1 3

5

6 3 2

x x x

x x x

6

Optimality Conditions for Minimization Problem

Basic Feasible Solution:

Optimality Conditions:

z xB xN1 xN2 RHS

z 1 0 cBB-1N1-cN1 cBB-1N2-cN2 z

xB 0 I B-1N1 B-1N2 b

1

2

0

0j j

j j

z c for all j N

z c for all j N

7

Handling Entering Variable

If the nonbasic variable entering the basis is at its lower bound, then its value is increased.

If the nonbasic variable entering the basis is at its upper bound, then its value is decreased.

The amount of increase or decrease is determined by the constraint that the values of all basic (and nonbasic) variables remains within their lower and upper bounds.

8

Example

1 2 3

1 2 3

1 2 3

1

2

3

2 4

. . 2 10

4

0 4

0 6

1 4

Minimize x x x

s t x x x

x x x

x

x

x

z x1 x2 x3 x4 x5 RHS

z 1 2 4 1 0 0 -1

x4 0 2 1 1 1 0 9

x5 0 1 1 -1 0 1 5

The initial simplex tableau:

Entering variable: x2 Leaving variable: x5

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Example (cont’d)

z x1 x2 x3 x4 x5 RHS

z 1 -2 0 5 0 -4 -21

x4 0 1 0 2 1 -1 4

x2 0 1 1 -1 0 1 5

Next tableau:

Entering variable: x3

Leaving variable: x2

10

Example (cont’d)

z x1 x2 x3 x4 x5 RHS

z 1 3 5 0 0 1 -26

x4 0 3 2 0 1 1 2

x3 0 -1 -1 1 0 -1 2

Next tableau:

Entering variable: x1

Leaving variable: x4

11

Example (cont’d)

z x1 x2 x3 x4 x5 RHS

z 1 0 3 0 -1 0 -28

x1 0 1 2/3 0 1/3 1/3 2/3

x3 0 0 -1/3 1 1/3 -2/3 8/3

The current solution is an optimal solution.

Final tableau: