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Supplement
E
Linear Programming
A. Basic Concepts
1.
Linear programming is an optimization process with several characteristics
a.
A single
objective function
states mathematically what is being maximized or
minimized.
b.
Decision variables
represent choices that the decision maker can control. Solving
the problem yields their optimal values based on the assumption that decision
variables are continuous.
c.
Constraints
are limitations that restrict the permissible choices for the decision
variables, which can be expressed mathematically in one of three ways:
≤
, =, or
≥
constraints.
d.
The
feasible region
includes all of the combinations of the decision variables
which satisfy the given constraints. Usually an infinite number of possible
solutions.
e.
A
parameter
, also known as a coefficient or given constant, is a value that the
decision maker cannot control and that does not change when the solution is
implemented.
f.
Assume parameters are known with certainty, and without doubt.
g.
The objective function and constraints are assumed to be
linear
, which implies
proportionality and additivity—there can be no products or powers of decision
variables.
h.
We assume the model to exhibit nonnegativity, which means that the decision
variables must be positive or zero.
2.
Formulating a problem
a.
Step 1:
Define the decision variables.
•
What must be decided?
•
Define each decision variable specifically, remembering that the definitions
must be equally useful in the constraints.
b.
Step 2:
Write out the objective function.
•
What is to be maximized or minimized?
•
Write out an objective function to make what is being optimized a linear
function of the decision variables.
c.
Step 3:
Write out the constraints.
•
What limits the values of the decision variables?
E1
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Supplement E: Linear Programming
•
Identify the constraints and the parameters for each decision variable in
them. To be formally correct, also write out the nonnegativity constraints.
d.
As a consistency check, make sure the same unit of measure is being used on
both sides of each constraint and the objective function.
3.
Problem Formulation.
Use
Application E.1: Crandon Manufacturing
.
The Crandon Manufacturing Company produces two principal product lines. One
is a portable circular saw, and the other is a precision table saw. Two basic
operations are crucial to the output of these saws:
fabrication and assembly. The
maximum fabrication capacity is 4000 hours per month; each circular saw
requires 2 hours, and each table saw requires 1 hour. The maximum assembly
capacity is 5000 hours per month; each circular saw requires 1 hour, and each
table saw requires 2 hours. The marketing department estimates that the
maximum market demand next year is 3500 saws per month for both products.
The average contribution to profits and overhead is $900 for each circular saw
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 Spring '09
 ahmed
 Management

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