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:
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: