Maximize z = c_{1}x_{1} + c_{2}x_{2} + c_{3}x_{3} + .........+ c_{n}x_{n}

subject to

a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3} + .........+ a_{1n}x_{n} ≤
b_{1}

a_{21}x_{1} + a_{22}x_{2} + a_{23}x_{3} + .........+ a_{2n}x_{n} ≤
b_{2}

.........................................................................

a_{m1}x_{1} + a_{m2}x_{2} + a_{m3}x_{3} + .........+ a_{mn}x_{n} ≤
b_{m}

x_{1}, x_{2},....., x_{n}≥
0

Where:

c_{j} (j = 1, 2, ...., n) in the objective function are called
the cost or profit coefficients.

b_{i} (i = 1, 2, ...., m) are called resources.

a_{ij} (i = 1, 2, ...., m; j = 1, 2, ...., n) are called technological
coefficients or input-output coefficients.

Introducing **slack variables** to convert inequalities to
equalities

a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3} + .........+ a_{1n}x_{n} + s_{1} = b_{1}

a_{21}x_{1} + a_{22}x_{2} + a_{23}x_{3} + .........+ a_{2n}x_{n} + s_{2} = b_{2}

..............................................................................

a_{m1}x_{1} + a_{m2}x_{2} + a_{m3}x_{3} + .........+ a_{mn}x_{n} + s_{m} = b_{m}

x_{1}, x_{2},....., x_{n}≥ 0

s_{1}, s_{2},....., s_{m }≥
0

An **initial basic feasible solution** is obtained by setting
x_{1} = x_{2} =........ = x_{n} = 0

s_{1} =
b_{1}

s_{2} =
b_{2}

..............

s_{m} =
b_{m}

The *initial simplex table* is formed by writing out the
coefficients and constraints of a LPP in a systematic tabular form.
The following table shows the structure of a simplex table.

c_{j} |
c_{1} |
c_{2} |
c_{3} |
--- | c_{n} |
||
---|---|---|---|---|---|---|---|

c_{B} |
Basic variables B |
x_{1} |
x_{2} |
x_{3} |
--- | x_{n} |
Solution values b (=X _{B}) |

c_{B1} |
x_{1} |
a_{11} |
a_{12} |
a_{13} |
--- | a_{1n} |
b_{1} |

c_{B2} |
x_{2} |
a_{21} |
a_{22} |
a_{23} |
--- | a_{2n} |
b_{2} |

c_{B3} |
x_{3} |
a_{31} |
a_{32} |
a_{33} |
--- | a_{3n} |
b_{3} |

--- | ----- | ---- | ---- | ----- | --- | ---- | ----- |

c_{Bm} |
x_{n} |
a_{m1} |
a_{m2} |
a_{m3} |
--- | a_{mn} |
b_{m} |

z_{j}-c_{j} |
z_{1}-c_{1} |
z_{2}-c_{2} |
z_{3}-c_{3} |
--- | z_{n}-c_{n} |

Where:

c_{j} = coefficients of the variables (m + n) in the objective
function.

c_{B} = coefficients of the current basic variables in the objective
function.

B = basic variables in the basis.

X_{B} = solution values of the basic variables.

z_{j}-c_{j} = index row.