Standard Form is the usual and most intuitive form of describing a linear programming problem
standard form can be described in 2 forms:
- equation form
- block matrix form
both represents the same thing
Equation Form
a linear programming problem in standard form is expressed in 3 parts (3 options to choose from):
- for options 2 and 3, use slack variables to convert to preferred option 1 form
|
PART |
OPTION 1 (preferred) |
OPTION 2 |
OPTION 3 |
|---|---|---|---|
|
objective linear function |
maximize or minimize Z = c1x1 + c2x2 + … + cnxn |
maximize Z = c1x1 + c2x2 + … + cnxn |
minimize Z = c1x1 + c2x2 + … + cnxn |
|
linear constraints |
a11x1 + a12x2 + … + a1nxn = b1 |
a11x1 + a12x2 + … + a1nxn ≤ b1 |
a11x1 + a12x2 + … + a1nxn ≥ b1 |
|
non-negative variable constraints |
x1 ≥ 0 | ||
Block Matrix Form
these 3 parts are usually expressed in matrix/vector form (3 options to choose from):
|
PART |
OPTION 1 |
OPTION 2 |
OPTION 3 |
|---|---|---|---|
|
objective linear function |
maximize or minimize Z = cTx |
maximize Z = cTx |
minimize Z = cTx |
|
linear constraints |
Ax = b |
Ax ≤ b |
Ax ≥ b |
|
non-negative variable constraints |
x ≥ 0 | ||
where:
- c - is a n-dimensional column vector
- x - is a n-dimensional row vector
- A - is a m-by-n matrix
- b - is a m-dimensional column vector