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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