Page 388 - A Course in Linear Algebra with Applications

372 Chapter Ten: Linear Programming

r
x
The condition on factory Fi is that 52 ij < ii while that on
m
x
s
warehouse Wj is 52 ij — j • We are therefore faced with the
following linear programming problem:

minimize: z = Y_, 2_. C-ij Xij
i = l j=l

f n
52 x^ < n, i = i,...,m
J'=l
subject to: < ™ .
x
s
/ J ij — ji 3 = *•J • • • > ^
i = l

The general linear programming problem
After these examples we are ready to describe the general
form of a linear programming problem.
Let xi, x 2, • • •, x n be variables. There is given a linear
function of the variables

z = ciXi + c 2x 2 H h c nx n,

called the objective function, which has to be maximized or
minimized. The variables Xj are subject to a number of linear
conditions, called the constraints, which take the form

anxi + a i2x 2 -\ h a inx n < or = or > bi,

i = 1, ,..., m. In addition, certain of the variables may be
2
constrained, i.e., they must take non-negative values. The gen-
eral linear programming problem therefore takes the following
form: