General Linear Programming Problem

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

Consider the following general mathematical formulation of LPP.

Optimize (maximize or minimize)
z = c₁x₁ + c₂x₂ + c₃x₃ + .........+ c_nx_n

subject to

a₁₁x₁ + a₁₂x₂ + a₁₃x₃ + .........+ a_1nx_n ( ≤, =,≥ ) b₁
a₂₁x₁ + a₂₂x₂ + a₂₃x₃ + .........+ a_2nx_n ( ≤, =,≥ ) b₂
................................................................................................
a_m1x₁ + a_m2x₂ + a_m3x₃ + .........+ a_mnx_n ( ≤, =,≥ ) b_m
x₁, x₂,....., 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_jx_j

subject to
a_ijx_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.