Applications Using Linear Programming

Here are the example videos promised on the last page. Again, don't expect to just watch these once. You may need to start and stop them, or watch them numerous times and write out notes while viewing them in order to learn this material. Also, understand that these problems are LONG, so each problem may take a while for the application problems in your homework assignment for 4.5. That is OK. It is to be expected for this material. There is not a short cut here! Sorry about that:-)

The good thing about this material is that it really is very applicable. Not that every one of you will be doing this again some day, but you will definitely be interacting with people that do this and with merchandise and salespeople and gadgets which were directly influenced, either consciously  or without knowledge, by this form of mathematics. Entire courses in engineering and mathematics are completely devoted to linear algebra and optimization, which is what is introduced in this section.

Nerdy people really do have a huge influence on the world, whether or not you choose to be one of them! At least please make some effort to give this section your best try, and you may be surprised once you go through several of these videos, as there is a pattern to this material. Here are the steps for all of these problems:

  1. Read the problem carefully. You will be reading the problem more than one time, as it contains lots of information. 
  2. Decide what the objective function is. This will be the quantity you are asked to find either the maximum or minimum (or both) for. Write a mathematical expression for the objective function. Do not forget to always define any variables you use in any problem.
  3. Write your system of constraints. This will be a system of inequalities that are either given to you, described in the problem, or that you need to know from the context of the problem. (For example, you can only make as little as zero of something , so if x represents the number of items made at a factory, then LaTeX: x\ge0\:x0would be one of the constraints, even though it may or may not be mentioned explicitly.)
  4. Graph the system of constraints on a single graph. The region where all of the shadings from the various inequalities in the constraints overlap is called the feasibility region.
  5. Find the vertices (corners) of the feasibility region. 
  6. List the vertices of the feasibility region, and find the value for the objective function at each of these corners. The maximum and minimum values of the objective function will be the maximum and minimum values of the objective function found in this step. The associated values of the coordinates of the vertex at the maximum or minimum of the objective function will be the values of each of the variables upon which the objective function depends when the objective function takes on its max or min. Sometimes these values are asked for as well as the actual maximum or minimum value for the objective function itself.
  7. Answer the question asked using a sentence, or possibly more than one sentence. Include units where applicable.

Please notice in these videos that even though each video represents a different presenters version of how to solve one of these problems, they all go through the steps above. Sometimes part of what is done is done verbally, for example the presenter may say what he or she is defining the variables to be, but all of the steps are included in each case.