Consider a company which produces two types of products and . Production of these products is supported by two workstations and , with each station visited by both product types. If workstation is dedicated completely to the production of product type , it can process 40 units per day, while if it is dedicated to the production of product , it can process 60 units per day. Similarly, workstation can produce daily 50 units of product and 50 units of product , assuming that it is dedicated completely to the production of the corresponding product. If the company's profit by disposing one unit of product is $200 and that of disposing one unit of is $400, and assumning that the company can dispose its entire production, how many units of each product should the company produce on a daily basis to maximize its profit?

**Solution:** First notice that this problem is an *
optimization* problem. Our *objective* is to *maximize* the
company's profit, which under the problem assumptions, is equivalent
to maximizing the company's *daily* profit. Furthermore, we are going to
maximize the company profit by adjusting the levels of the daily
production for the two items and . Therefore, these daily
production levels are the control/decision factors, the values of
which we are called to determine. In the analytical formulation of the
problem, the role of these factors is captured by modeling them as the
problem *decision variables*:

- := number of units of product to be produced daily
- := number of units of product to be produced daily

Equation 1 will be called the *objective function* of
the problem, and the coefficients 200 and 400 which multiply the
decision variables in it, will be called the *objective function
coefficients*.

Furthermore, any decision regarding the daily production levels for
items and in order to be realizable in the company's
operation context must observe the production capacity of the two
worksations and . Hence, our next step in the problem
formulation seeks to introduce these *technological* constraints
in it. Let's focus first on the constraint which expresses the finite
production capacity of workstation . Regarding this constraint,
we know that one day's work dedicated to the production of item
can result in 40 units of that item, while the same period dedicated
to the production of item will provide 60 units of it. *Assuming
that production of one unit of product type , requires a
constant amount of processing time at workstation
*, it follows that: and . *Under the further assumption that the combined
production of both items has no side-effects, i.e., does not impose any
additional requirements for production capacity of workstation
(e.g., zero set-up times)*, the total capacity (in terms of time
length) required for producing units of product and
units of product is equal to . Hence, the technological constraint imposing the
condition that our total daily processing requirements for workstation
should not exceed its production capacity, is analytically
expressed by:

Notice that in Equation 2 time is measured in days.

Following the same line of reasoning (and under similar assumptions), the constraint expressing the finite processing capacity of workstation is given by:

Constraints 2 and 3 are known as the *
technological constraints* of the problem. In particular, the
coefficients of the variables in them,
, are known as the *technological
coefficients* of the problem formulation, while the values on the
right-hand-side of the two inequalities define the *right-hand
side (rhs)* vector of the constraints.

Finally, to the above constraints we must add the requirement that any permissible value for variables must be nonnegative, i.e.,

since these values express production levels. These constraints are
known as the variable *sign restrictions*.

Combining Equations 1 to 4, the analytical formulation of our problem is as follows:

s.t.

Fri Jun 20 15:03:05 CDT 1997