Today I want to begin introducing you to a powerful optimization technique you might have occasion to use in your business. The technique is called Linear Programming and is the primary technique taught and used in Operations Research. Wikipedia defines Operations Research as "a discipline that deals with the application of advanced analytical methods to help make better decisions. It is often considered to be a sub-field of mathematics. The terms management science and decision science are sometimes used as synonyms".

Linear Programming is a fairly difficult technique to master because it involves some advanced math and the ability translate a business problem into a set of equations representing what it is you want to optimize (i.e., maximize or minimize the value of) and the constraints that exist upon how you can solve the problem (i.e., subject to constraints on labor, capital, machinery, time, etc...). The ability to translate a business problem into a set of equations is generally acquired by studying standard types of business problems that Linear Programming has been applied to and then using those example solutions as a template for applying the technique to your own similar situation.

In today's blog, I want to simply display the main set of formulas that are used in linear programming. I am reproducing the main linear programming formulas from the book Mathematical Programming for Agricultural, Enviornmental, and Resource Economics by Harry M. Kaiser & Kent D. Messer, Wiley, 2012. This was also an opportunity for me to take the MathJax library I setup yesterday for another test drive and learn more about how to add some professional looking math to my blog.

Here is the general form of the Linear Program (LP) model. First we need to define the objective function $Z$ that we want to maximize or minimize.

\[Z = c_1x_1 + c_2x_2 + ... + c_nx_n\]

Then we need to specify the various business constraints (labor costs, material costs, transport costs) that our business decision is subject to. These constraints are formulated in terms of equations with a left hand side being less than, equal to, or greater than some value in the right hand side:

\[a_{11}x_1\ + a_{12}x_2 + \cdots + a_{1n}x_n \lbrace {\le, =, \ge} \rbrace b_1\]

\[a_{21}x_1\ + a_{22}x_2 + \cdots+ a_{2n}x_n \lbrace {\le, =, \ge} \rbrace b_2\]

\[\cdots\]

\[\cdots\]

\[a_{m1}x_1\ + a_{m2}x_2 + \cdots+ a_{mn}x_n \lbrace {\le, =, \ge} \rbrace b_m\]

Finally, we generally add the constraint that all "activities" are non-negative (we can't have negative activity values in our production model).

\[x_1, x_2, \cdots x_n \ge 0\]

We'll explore the technique of linear programming in further detail in my next blog

Note: If you want to see the tex code used to generate the equations in this blog, all you have to do is right-click on the equation and MathJax supplies a viewer application for inspecting and copying the formulas.