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66 CHAPTER 2 AN INTRODUCTION TO LINEAR PROGRAMMING

Can you recognize Because unbounded solutions cannot occur in real problems, the first thing you
whether a linear should do is to review your model to determine whether you have incorrectly
programme involves
alternative optimal formulated the problem. In many cases, this error is the result of inadvertently
solutions, infeasibility or omitting a constraint during problem formulation.
is unbounded? Try
Problems 17 and 18

NOTES AND COMMENTS

1 Infeasibility is independent of the objective a change in the objective function may cause a
function. It exists because the constraints are so previously unbounded problem to become
restrictive that there is no feasible region for the bounded with an optimal solution. For example,
linear programming model. Thus, when the graph in Figure 2.20 shows an unbounded
you encounter infeasibility, making changes in the solution for the objective function Max 20X +10Y.
coefficients of the objective function will not help; However, changing the objective function to
the problem will remain infeasible. Max 20X 10Y will provide the optimal
2 The occurrence of an unbounded solution is solution X ¼ 2and Y ¼ 0eventhoughno
often the result of a missing constraint. However, changes have been made in the constraints.

2.7 General Linear Programming Notation

In this chapter we showed how to formulate linear programming models for the
GulfGolf and M&D Chemicals problems. To formulate a linear programming
model of the GulfGolf problem we began by defining two decision variables: S ¼
number of standard bags, and D ¼ number of deluxe bags. In the M&D Chemicals
problem, the two decision variables were defined as A ¼ number of litres of product
A, and B ¼ number of litres of product B. We selected decision-variable names of S
and D in the GulfGolf problem and A and B in the M&D Chemicals problem to
make it easier to recall what these decision variables represented in the problem.
Although this approach works well for linear programmes involving a small number
of decision variables, it can become difficult when dealing with problems involving a
large number of decision variables.
A more general notation that is often used for linear programmes uses the letter x
with a subscript. For instance, in the GulfGolf problem, we could have defined the
decision variables as follows:

x 1 ¼ number of Standard bags
x 2 ¼ number of Deluxe bags

In the M&D Chemicals problem, the same variable names would be used, but their
definitions would change:

x 1 ¼ number of litres of product A
x 2 ¼ number of litres of product B
A disadvantage of using general notation for decision variables is that we are no
longer able to easily identify what the decision variables actually represent in the
mathematical model. However, the advantage of general notation is that formulating

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