Lecture 6
Linear programming
Source:
/1
/
D.R.Anderson, D.J.Sweeney, T.A.Williams. Quantitative Methods for Business,
SouthWestern College Publishing, 11th edition.
Chapter 7.
Contents:
6.1
A simple maximization problem
6.2
Graphical solution procedure
6.3
A simple minimization problem
Linear programming
is a problemsolving approach developed to help managers
make decisions. This approach is based on finding an extreme value of some
linear
function
(
objective function
) subject to certain
constraints
in the form of
linear
inequalities
.
We are either trying to maximize or minimize our objective function.
That is why
these linear programming problems are classified as maximization or minimization
problems, or just optimization problems.
In this Lecture, we will do problems that involve
only two variables, and therefore, can be solved by graphing. We begin by solving a
maximization problem.
6.1 A simple maximization problem
Example 6.1
RMC, Inc. produces chemicalbased products. Three raw materials
are used to produce two products: a fuel additive and a solvent base. The three raw
materials are blended to form the fuel additive and solvent base as indicated in
Table 6.1
.
It shows that a ton of fuel additive is a mixture of 0.4 ton of material 1 and 0.6 ton of
material 3. A ton of solvent base is a mixture of 0.5 ton of material 1, 0.2 ton of material 2,
and 0.3 ton of material 3.
Table 6.1 Material requirements per ton for the RMC problem
Product
Material
Fuel Additive
Solvent Base
Material 1
0.4
0.5
Material 2
0.2
Material 3
0.6
0.3
RMC's production is constrained by a limited availability of the three raw materials. For
the current production period, RMC has available the following quantities of each raw
material.
Material
Amount Available for Production
Material 1
20 tons
Material 2
5 tons
Material 3
21 tons
Because of spoilage and the nature of the production process, any materials not used for
current production are useless and must be discarded.
The accounting department analyzed the production figures, assigned all relevant costs,
and arrived at prices for both products that will result in a profit contribution of $40 for
every ton of fuel additive produced and $30 for every ton of solvent base produced.
Problem
: determine the number of tons of fuel additive and the number of tons of solvent
base to produce in order to
maximize
total profit
.
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View Full DocumentWe begin with the mathematical statement or
mathematical model
of this problem.
To develop a mathematical model for a linear programming problem we should
1.
Describe the Objective.
RMC's objective is to maximize the total profit.
2.
Describe Each Constraint.
Three constraints limit the number of tons of fuel additive and the number of tons of
solvent base that can be produced.
Constraint
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 Spring '11
 dd
 Linear Programming, Optimization, objective function, Constraint, feasible region, Rmc

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