MSE 331 Introduction to optimization

Upon completion of MSE 331, students are able to:

  • Formulate linear and integer programs to model real-life problems
  • Solve linear programs using the graphical solution and simplex methods
  • Formulate and interpret the dual of a linear program
  • Implement sensitivity analyses to account for uncertainties
  • Collect data to determine or estimate the values of problem parameters
  • Use Microsoft Excel to solve linear and integer programs

What is optimization?

Optimization is the selection of the best element (with regard to some criteria) from some set of available alternatives. Another name for optimization is mathematical programming.

Example

A company manufactures two products A and B and the profit per unit sold is $3 and $5, respectively. Each product has to be assembled on a particular machine, each unit of product A takes 3 hours of assembly time and each unit of product B takes 2 hours of assembly time. The company estimates that the effective daily working hour of the machine used for assembly is 18 hours. At least 2 units of product A and 1 unit of product B must be produced daily.

Q: What is the optimum number of products A and B to produce daily to maximize total profit?

To solve this type of problem, we model it mathematically, as follows:

Let x1 and x2 be the number of product A and B to be produced, respectively. We call x1 and x2decision variables.

We then seek to maximize the objective function Z = 3x1 + 5x2

This maximization is subject to a number of constraints:

3x1 + 2x2 <= 18

x1>= 2

X2 >= 1

What is mathematical modelling?

You may have noticed that in order to solve it, the problem above had to be first modelled as a series of mathematical equations. Mathematical modelling is a powerful tool that transforms messy real-life problems into models that can be solved mathematically. The solutions can be used by management to make important decisions for organizations.
High-level view of mathematical modelling: Management involves a real-life problem, decisions, a mathematical model, and a solution, mixed in with assumptions and methodology.

What is operations research?

Optimization is just one of many methodologies used in the field of operations research.

Operations research is a discipline that deals with the application of advanced analytical methods to help make better decisions. It is a rational, structured, systems approach to problem solving.
Related and sometimes overlapping fields to operations research are operational research, management science, analytics, decision sciences, and industrial engineering.

Applications of operations research

Operations research can be applied in many fields, including finance, supply chain, health care, and more.

Finance

  • Portfolio investments
  • Strategies for pension and insurance
  • Revenue management

Supply chain

  • Production and inventory planning
  • Location analysis
  • Job sequencing
  • Vehicle routing

Health care

  • Operating room scheduling
  • Nurse scheduling
  • ER waiting rooms
  • Organ allocation
  • Managing blood inventory

Sports

  • Scheduling
  • Tactics and strategies
  • Forecasting
....and much more!