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Mathematical Modeling Process For Real Situations

1798 words - 8 pages

In general, experimenting with models requires less time and is less expensive than experimenting
with the real object or situation. A model airplane is certainly quicker and less
expensive to build and study than the full-size airplane. Similarly, the mathematical model
in equation (1.1) allows a quick identification of profit expectations without actually requiring
the manager to produce and sell x units. Models also have the advantage of reducing the
risk associated with experimenting with the real situation. In particular, bad designs or bad
decisions that cause the model airplane to crash or a mathematical model to project a
$10,000 loss can be avoided in the real situation.
The value ...view middle of the document...

A production capacity constraint would be
necessary if, for instance, 5 hours are required to produce each unit and only 40 hours of
production time are available per week. Let x indicate the number of units produced each
week. The production time constraint is given by
8 Chapter 1 Introduction
Herbert A. Simon, a Nobel
Prize winner in economics
and an expert in decision
making, said that a
mathematical model does
not have to be exact; it just
has to be close enough to
provide better results than
can be obtained by common
sense.
5x … 40 (1.2)
The value of 5x is the total time required to produce x units; the symbol indicates that the
production time required must be less than or equal to the 40 hours available.
The decision problem or question is the following: How many units of the product
should be scheduled each week to maximize profit? A complete mathematical model for
this simple production problem is
The x 0 constraint requires the production quantity x to be greater than or equal to
zero, which simply recognizes the fact that it is not possible to manufacture a negative
number of units. The optimal solution to this model can be easily calculated and is given by
x 8, with an associated profit of $80. This model is an example of a linear programming
model. In subsequent chapters we will discuss more complicated mathematical models and
learn how to solve them in situations where the answers are not nearly so obvious.
In the preceding mathematical model, the profit per unit ($10), the production time per
unit (5 hours), and the production capacity (40 hours) are environmental factors that are not
under the control of the manager or decision maker. Such environmental factors, which can
affect both the objective function and the constraints, are referred to as uncontrollable
inputs to the model. Inputs that are controlled or determined by the decision maker are
referred to as controllable inputs to the model. In the example given, the production quantity
x is the controllable input to the model. Controllable inputs are the decision alternatives specified
by the manager and thus are also referred to as the decision variables of the model.
Once all controllable and uncontrollable inputs are specified, the objective function
and constraints can be evaluated and the output of the model determined. In this sense,
the output of the model is simply the projection of what would happen if those particular
5x … 40
x Ú 0 f constraints
Maximize
subject to (s.t.)
P = 10x objective function
environmental factors and decisions occurred in the real situation. A flowchart of how
controllable and uncontrollable inputs are transformed by the mathematical model into
output is shown in Figure 1.4. A similar flowchart showing the specific details of the production
model is shown in Figure 1.5.
As stated earlier, the uncontrollable inputs are those the decision maker cannot influence.
The specific controllable and uncontrollable inputs of a model depend on the...

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