Hidden in our models of these problems are a number of assumptions. The usefulness of a model is directly related to how close reality matches up with these assumptions.

The first assumption is due to the linear form of our functions.
Since the objective is linear, the contribution to the objective of
any decision variable is proportional to the value of the decision
variable. Producing twice as much of a product produces twice as much
profit; buying twice as many pages of ads costs twice as much. This
is the *Proportionality Assumption*.

Furthermore, the contribution of a variable to the objective is
independent of the values of the other variables. One notebook
computer is worth $750, independent of how many desktop computers we
produce. This is the *Additivity Assumption*.

Similarly, since each constraint is linear, the contribution of each variable to the left hand side of each constraint is proportional to the value of the variable and independent of the values of any other variable.

These assumptions are quite restrictive. We will see, however, that clever modeling can handle situations that may appear to violate these assumptions.

The next assumption is the *Divisibility Assumption*: it is
possible to take any fraction of any variable. Rethinking the
Marketing example, what does it mean to purchase 2.67 television ads?
It may be that the divisibility assumption is violated in this
example. Or, it may be that the units are such that 2.67 ``ads''
actually corresponds to 2666.7 minutes of ads, in which case we can
``round off'' our solution to 2667 minutes with little doubt that we
are getting an optimal or nearly optimal solution. Similarly, a
fractional production quantity may be worisome if we are producing a
small number of battleships or be innocuous if we are producting
millions of paperclips. If the Divisibility Assumption is important
and does not hold, then a technique called *integer programming*
rather than linear programming is required. This technique takes
orders of magnitude more time to find solutions but may be necessary
to create realistic solutions. You will learn more about this in
45-761.

The final assumption is the *Certainty Assumption*: linear
programming allows for no uncertainty about the numbers. An ad will
reach the given number of people; the number of assembly hours we
give will certainly be available.

It is very rare that a problem will meet all of the assumptions exactly. That does not negate the usefulness of a model. A model can still give useful managerial insight even if reality differs slightly from the rigorous requirements of the model. For instance, the knowledge that our chip inventory is more than sufficient holds in our first model even if the proportionality assumptions are not satisfied completely.

Mon Aug 24 16:30:59 EDT 1998