# 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 + .........+ cnxn

subject to

a11x1 + a12x2 + a13x3 + .........+ a1nxn ( ≤, =,≥ ) b1
a21x1 + a22x2 + a23x3 + .........+ a2nxn ( ≤, =,≥ ) b2
................................................................................................
am1x1 + am2x2 + am3x3 + .........+ amnxn ( ≤, =,≥ ) bm
x1, x2,....., xn ≥ 0

"Mathematical structures are among the most beautiful discoveries made by the human mind" - Douglas Hofstadter.

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 = cjxj

subject to
aijxj (≤, =,≥ ) bi; i = 1, 2, ....., m (constraints)

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

The ∑ summation symbol considerably reduces the amount of writing lengthy expressions.

Where all cj's, aij's, bi's are constants and xj'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 xj ≥ 0 means that the xj's must be non-negative.

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