Linear Programming
Graphical Method
• Linear Programming – a branch of applied Mathematics, which
is a mathematical technique that involves maximizing and
minimizing a linear function subject to given linear constraints
The term “linear” refers to the relationship involving two or more
variables, which show first degree mathematical statement. The
term “programming” refers to the use of certain mathematical
techniques or algorithms to obtain best possible solution or the
OPTIMAL solution.
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Characteristics of Linear
Programming Problems
• Two parts:
- Objective function – is
a mathematical statement
reflecting the objective of the operation. A single quantifiable
objective must be specified by the decision maker. The objective
of the decision maker must be to maximize or minimize.
- The decision maker must achieve the objective of the problem
and must not violate the limitations or constraints. These
“constraints” are referred to the availability of resources like
labor time, machine time, raw materials, work or storage space,
etc. These resources must be limited to supply. Two kinds of
constraints in LPP, the explicit and the implicit.
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Linear programming Problems
• Simple Linear programming Problems
• Duality and sensitivity Analysis
• Transportation and Assignment
Problems
• Integer programming
• Goal Programming
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Linear Programming Model or Set up
• Objective function: Maximize
Subject to constraints
• Objective Function: Minimize
Subject to constraints
where: a’s, b’s and c’s are numerical values called parameter and
x’s are the decision variables that represent the level of activity in
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the problem
Methods of solving Simple
linear Programming Problems.
• Graphical Method or Geometric Method
• Simplex Method or tabular Method
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Possible Solutions of LPP
• Feasible – the solution is said to be feasible if the
graphical solutions of the problem constraints have a
common intersection, called the feasible region.
the feasible region may be bounded or unbounded;
bounded if the region is a closed plane figure, and
unbounded if the region is an open plane figure.
• Infeasible – an LPP has infeasible solutions if the
problem constraints do not intersect, that is, no
feasible region is formed.
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Examples:
Solve the LPP by graphical method
1.
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Examples:
• Solve the LPP by graphical method
2.
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Formulation of LP models
• Analyze and interpret the problem
• Determine the decision variables in the problem. These
variables correspond to the decision that must made in order to
identify a solution to the problem.
• Present and summarize the data in tabular form (if possible).
• Identify the objective of the problem (to maximize or to
minimize). Translate the objective of the problem to a
mathematical statement (this is the objective function).
• Identify the limitations or restrictions in the problem and
represent them as linear expressions involving the decision
variables. Words or expressions to denote are “no more than”,
“available”, “at most”, “limited to”, etc.
• Gather pertinent data or make appropriate approximations for all
arbitrary values in the problem.
• Form the model.
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LPP and their Corresponding
Models
• Production Problem
the GEM maker of jewelry makes two bracelet designs,
heart designs and flower design. The bracelets are made of gold
and platinum. The store has 28 ounces of gold and 20 ounces of
platinum. Each heart design bracelet requires 3 ounces of gold
and 2.5 ounces of platinum and makes a profit of P2,500, while
each flower design bracelet requires 4.5 ounces of gold and 3
ounces of platinum and makes a profit of P 3,400. How many
heart design bracelets and flower design bracelets should be
produced to maximize the profit?
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LPP and their Corresponding
Models
• Project Mix Problem
The Doodle food product company makes instant noodles
from several ingredients. Three of the ingredients, flour,
squash, and mixed vegetables provide vitamin A, B and C. The
company wants to know how many grams of flour, squash and
mixed vegetables should include in each pack to meet the
minimum requirements of 240 mg of vitamin A, 198 mg of
vitamin B and 135 mg of vitamin C while minimizing cost. The
following table shows the information on the vitamin content of
each gram of the ingredients.
Ingredients
Vitamin A(mg)
Flour
5
Squash
3
2
Mixed vegetables
2
3
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Vitamin B(mg) Vitamin C(mg)
2
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