Linear Programming Lecture Notes - Penn State Personal Web Server

Remark 3.16. We can deal with constraints of the form:
(3.21) ai1x1 + ai2x2 + · · · + ainxn ≥ bi
in a similar way. In this case we subtract a surplus variable si to obtain:
ai1x1 + ai2x2 + · · · + ainxn − si = bi
Again, we must have si ≥ 0.
Theorem 3.17. Every linear programming problem in standard form can be put into
canonical form.
Proof. Recall that Ax = b if and only if Ax ≤ b and Ax ≥ b. The second inequality
can be written as −Ax ≤ −b. This yields the linear programming problem:
⎧
max z(x) =c
⎪⎨
(3.22)
⎪⎩
T x
s.t. Ax ≤ b
− Ax ≤ −b
x ≥ 0
Defining the appropriate augmented matrices allows us to convert this linear programming
problem into canonical form.
Exercise 24. Complete the “pedantic” proof of the preceding theorem by defining the
correct augmented matrices to show that the linear program in Expression 3.22 is in canonical
form.
The standard solution method for linear programming models (the Simplex Algorithm)
assumes that all variables are non-negative. Though this assumption can be easily relaxed,
the first implementation we will study imposes this restriction. The general linear programming
problem we posed in Expression 3.15 does not (necessarily) impose any sign restriction
on the variables. We will show that we can transform a problem in which xi is unrestricted
into a new problem in which all variables are positive. Clearly, if xi ≤ 0, then we simply
replace xi by −yi in every expression and then yi ≥ 0. On the other hand, if we have the
constraint xi ≥ li, then clearly we can write yi = xi − li and yi ≥ 0. We can then replace xi
by yi + li in every equation or inequality where xi appears. Finally, if xi ≤ ui, but xi may
be negative, then we may write yi = ui − xi. Clearly, yi ≥ 0 and we can replace xi by ui − yi
in every equation or inequality where xi appears.
If xi is unrestricted in sign and has no upper or lower bounds, then let xi = yi − zi where
yi, zi ≥ 0 and replace xi by (yi − zi) in the objective, equations and inequalities of a general
linear programming problem. Since yi, zi ≥ 0 and may be given any values as a part of the
solution, clearly xi may take any value in R.
Exercise 25. Convince yourself that the general linear programming problem shown
in Expression 3.15 can be converted into canonical (or standard) form using the following
steps:
(1) Every constraint of the form xi ≤ ui can be dealt with by substituting yi = ui − xi,
yi ≥ 0.
(2) Every constraint of the form li ≤ xi can be dealt with by substituting yi = xi − li,
yi ≥ 0.
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