LINEAR PROGRAMMING: Minimization
Introduction
In the previous lesson we have discussed what linear programming problems are
and how they can be formulated. We have also learned how to find the optimal solution
to a maximization problem using the graphical solution method for a twovariable case or
the simplex procedure that can handle any number of variables.
In many business settings, however, an analyst is interested in minimizing the
value of his objective function. For instance, in a diet problem, dietician’s maybe to
minimize the cost of foods subject to minimum daily requirements. The computational
procedures discussed in the preceding lesson are readily applicable to a minimization
problem. In this lesson we will show how the simplex method can be used to find the
optimal solution to a problem whose main objective is to minimize costs.
Minimization Problem
As we pointed out earlier, the criterion for optimality of an objective function of a
linear program may be its minimization rather than maximization. Usually a cost
minimization objective function is associated with a set of greaterthanorequalto
constraints or equality constraints. Such problems need to be modified and put into an
appropriate form before the simplex method can be applied for a solution. However, if
lessthanorequalto constraints occur with a minimization problem the application of the
simplex computational procedure is rather straight forward. We will first consider the
latter format of the problem for illustration and then discuss the former.
Suppose we are given the following LP problem to solve:
M
i
n
i
m
i
z
e
Z
=

4
x
1
+
5 x
2
Subject to
x
1
+
3 x
2
≤
5
2
x
1

3 x
2
≤
8
x
1,
x
2
≥
0
There are two ways in which we can solve this problem using the simplex method. The
first requires that we reserve the rule of introducing a new variable into the basis. Recall,
for maximization, the simplex rule requires that we select a variable with the most
negative objective function coefficient to enter the new solution. This is because the most
negative coefficient indicates the highest contribution to profit per unit of that variable.
An optimal solution is available when all the objective function coefficients are non
negative (zero or positive). For the same reason, a positive objective function coefficient
in a minimization problem indicates reduction in the cost per unit of that variable. In
order to minimize the cost, we would therefore select a variable for the next solution that
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View Full Documenthas the most positive coefficient. The process will terminate when all the objective
function coefficients are nonpositive (zero or negative). When this condition occurs we
have a solution to the minimization problem.
Thus, letting s
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 Spring '08
 Chin
 Linear Programming, Optimization, objective function

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