There are different ways to write a **general linear programming problem.**

Consider the following *general mathematical formulation of LPP***.**

Optimize (maximize or minimize)

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

If you have never taken a statistics course, then you will probably
find the following **∑** notation
strange, and perhaps even puzzling. To properly understand the text,
read the text atleast twice.

In ∑ notation, LPP can be written as

**Optimize (maximize or minimize) z** = c_{j}x_{j}

subject to

a_{ij}x_{j} (≤, =,≥ ) b_{i}; i = 1, 2,
....., m (constraints)

x_{j }≥
0; j = 1, 2, ....., n (non-negative restrictions)

Where all c_{j}'s, a_{ij}'s, b_{i}'s are constants
and x_{j}'s are decision variables. The expression (≤,
=,≥ ) means
that each constraint may take only one of the three possible forms:

- less than or equal to (≤)
- equal to (=)
- greater than or equal to (≥)

The expression x_{j }≥
0 means that the x_{j}'s must be non-negative.