Mastering Linear Programming Model Formulation

The Art of Linear Programming Model Formulation

Formulating a Linear Programming Model means selecting the elements of a major system problem and identifying how they relate. This is not an easy task in the case of real problems and includes tests for common trial and error. In fact, it is more of an art than a systematic procedure. However, there are steps that have proven useful in formulating linear programming models. These steps are:

• a. Set in verbal terms the objective to be achieved with the resolution of the problem. Select only one objective, such as "reduce costs" (minimize) or "increase contribution to profit" (maximize).
• b. Make a list of decisions that influence the achievement of this objective, as specific as possible.
• c. Make a list of the restriction factors that affect these decisions. Try to be accurate and complete.

General Types of Restrictions

Below is a list of several general types of restrictions. See if your problem has any of these conditions. Note also that there may be other restrictions. Usually, a problem does not include all types of constraints.

• Capacity constraints and/or resource availability: They are limits that are due to system limitations in terms of the amount of equipment, space, funding, raw materials, and manpower available. An example would be the constraint that refers to the land available for crops. These constraints are expressed as limitations, or inequalities of the type (≤). What we deal with as a resource cannot be greater than what we have available.
• Restrictions on market: There are limits (lower, upper, or both) on the amount of product that can be sold or used. For example, the maximum and historical minimum sales for a product. The latter would be a requirement or an inequality of the type (≥), because if there are committed sales for a certain product, I cannot decide to produce less than that amount because I could not fulfill my commitments to delivery. The first is a constraint or inequality of the type (≤), meaning I should not produce more product than what has historically been sold each season.
• Restrictions on quality or composition of a mixture: These restrictions limit the mixture of ingredients that usually define the quality of products.