LrChVIISIILinearProgramming052 - Qualitative Choice...

Qualitative Choice Analysis Workshop 1
Econometrics Laboratory

7.2 Linear Programming
Dr. Raja Mohammad Latif
Objective: To state the nature of a linear
programming problem, to introduce
terminology associated with it, and to solve
it geometrically.

Dr. Raja Latif, Math 131-052 (Feb 12-10June, 2006) 3
1 Chapter 7: Linear Programming. 7.2:
LINEAR PROGRAMMING
Abstarct: We will learn to state the nature of a linear pro-
gramming problem along with the introduction of terminology
associated with it, and then developing a method for its solution
geometrically.
Many business and economic problems are concerned
with optimizing (maximizing or minimizing) a function subject
to a system of equalities or inequalities. The function to be
optimized is called the objective function . Pro°t functions and
cost functions are examples of objective functions. The system
of equalities and inequalities to which the objective function
is subjected re±ects the constraints (for example, limitations
on resources such as materials and labor) imposed on the
solution(s) to the problem. Problems of this nature are called
mathematical programming problems . In particular, problems
in which both the objective function and the constraints are
expressed as linear equations or inequalities are called linear
programming problems.
A linear programming problem consists of a linear
objective function to maximized or minimized subject to certain
Department of Mathematics, KFUPM

Dr. Raja Latif, Math 131-052 (Feb 12-10June, 2006) 4
constraints in the form of linear equalities or inequalities.
Existence of a Solution. Consider a linear programming
problem with the set R of feasible points and objective function
z = Ax + BY:
1 :
If
R
is bounded, then
z
has a maximum and a minimum
value on
R:
2 :
If
R
is unbounded and
A ° 0 ; B ° 0 ;
and the constraints
include x ° 0 and y ° 0 ; then z has a minimum value on R but
not a maximum.
3
. If
R
is the empty set, then the linear programming
problem has no solution and z has neither a maximum nor a
minimum value.
Fundamental Theorem of Linear Programming
If a linear programming problem has a solution, it is
located at a corner point of the set of feasible points. If a linear
programming problem has multiple solutions, at least one of them
is located at a corner point of the set of feasible points. In either
case the corresponding value of the objective function is unique.