L
LINEAR PROGRAMMING PROBLEM
Let xi = decision variable for ith variable.
ci = profit or cost co-efficient of ith variable.
Z = function to be maximized or minimized.
Thus for n decision variables, the objective function is to maximize or minimize
Z = c1 x1 + c2 x2 + ... + cn xn
Let aij = co-efficient of the jth constraint and ith variable
bi = resource limitation for ith constraint
Thus the restrictions may be expressed in the general form
a11x1 + a12x2 + ... + a1nxn < b1
a21x1 + a22x2 + ... +a2nxn < b2
: : : :
am1x1 + am2x2 + ...+ amnxn < bm
and xi > 0 for all values of i from 1 to n
The same linear programming problem can be expressed in a more condensed form using summation
notation or matrix equation.
Maximize
1
n
ii
i
Z c x
Subject to
n
ij j
ji
ax
for all i = 1, 2, ..., m
and xj > 0 for all j = 1, 2, ..., n
or
Maximize Z = CX , subject to AX < B
where C is a row vector and X and B are column vectors. A is a co-efficient matrix of the order m x n.
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