Lecture 13 Introduction to linear
programming
Graphical Solution to LP Problems
Lecture Outline
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Model Formulation
Graphical Solution Method
Linear Programming Model
Solution
Solving Linear Programming Problems with
Excel
Sensitivity Analysis
Historical Note
In the World War II, when the war operations had to be
planned to economize expenditure, maximize damage to
the enemy, linear programming problems came to the
forefront.
The first problem in linear programming was formulated in
1941 by the Russian mathematician, L. Kantorovich and the
American economist, F. L. Hitchcock, both of whom worked
at it independently of each other. This was the well known
transportation problem.
In 1945, an English economist, G. Stigler, described yet
another linear programming problem – that of determining
an optimal diet.
Historical Note
In 1947, the American economist, G. B. Dantzig suggested an
efficient method known as the simplex method which is an
iterative procedure to solve any linear programming
problem in a finite number of steps.
L. Kantorovich and American mathematical economist, T. C.
Koopmans were awarded the Nobel prize in the year 1975
in economics for their pioneering work in linear
programming. With the advent of computers and the
necessary software, it has become possible to apply linear
programming model to increasingly complex problems in
many areas.
Introduction
Linear Programming (LP)
A model consisting of linear relationships
representing a firm’s objective and resource
constraints
LP is a mathematical modeling technique used to
determine a level of operational activity in order to
achieve an objective, subject to restrictions called
constraints
Types of LP
Types of LP (cont.)
Types of LP (cont.)
LP Model Formulation
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Decision variables
◦ mathematical symbols representing levels of activity of
an operation
Objective function
◦ a linear relationship reflecting the objective of an
operation
◦ most frequent objective of business firms is to maximize
profit
◦ most frequent objective of individual operational units
(such as a production or packaging department) is to
minimize cost
Constraint
◦ a linear relationship representing a restriction on
decision making
The Linear Programming Model (1)
The Linear Programming Model (2)
Examples of LP Problems (1)
Examples of LP Problems (2)
Examples of LP Problems (3)
Examples of LP Problems (4)
Examples of LP Problems (5)
Developing LP Model
Developing LP Model
Developing LP Model
Developing LP Model
Developing LP Model
Developing LP Model
Graphical Solution to LP Problems
The problems in this lecture contain no more than two
variables, and we will therefore be able to solve them
graphically in the xy-plane.
Recall that the solution set to a system of inequalities is
the region that satisfies all inequalities in the system. In
linear programming problems, this region is called the
feasible set, and it represents all possible solutions to the
problem.
Each vertex of the feasible set is known as a corner point.
The optimal solution is the point that maximizes or
minimizes the objective function, and the optimal value
is the maximum or minimum value of the function.
Graphical Solution to LP Problems
The context of a problem determines whether we want to
know the objective function’s maximum or the minimum
value.
If a linear programming problem represents the amount of
packaging material used by a company for their products,
then a minimum amount of material would be desired. If a
linear programming problem represents a company’s
profits, then a maximum amount of profit is desired. In
most of the examples in this section, both the maximum
and minimum will be found.
Graphical Solution to LP Problems
Fundamental Theorem of Linear Programming
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems
Graphical Solution to LP Problems