Applied Mathematics

Which mathematical formulations make a business decision explicit enough to be explained, tested and improved?

Many operational decisions are made by habit or by an opaque score. Without variables, an objective and constraints written down, nobody can say why a choice was made or what would change it.

A model written as variables, an objective and constraints turns an argument about opinions into a calculation that can be inspected. It also separates what the policy forbids (constraints) from what the business prefers (objective).

x is the decision, X the set of allowed decisions, f the objective and g_i the constraints with limits b_i.

1. State the decision variables and what the policy forbids.

2. Choose the simplest model that captures the trade-off (linear, integer, probabilistic).

3. Estimate parameters from data and mark every one that is only illustrative.

4. Check sensitivity: which parameter, if wrong, changes the decision?

1. Business question

2. Variables

3. Objective

4. Constraints

5. Solver

6. Explanation of the result

No published experiments. The Mathematical Optimization lab solves a two-variable linear program exactly, by vertex enumeration.

Mathematical Optimization

No original research results are claimed. This area is presented as a technical perspective with a computable demonstration on synthetic data.

• A model is only as good as its parameters; illustrative numbers prove nothing.
• Linear models cannot capture increasing or decreasing returns.
• Constraints that are not written down are not enforced.

Background reading by other authors. These are not Menlo Tech publications, and no Menlo Tech paper is cited because none has been published.

• Bertsimas, D. & Tsitsiklis, J. Introduction to Linear Optimization. Athena Scientific, 1997.
• Boyd, S. & Vandenberghe, L. Convex Optimization. Cambridge University Press, 2004.