Optimization Methods for Decision Making

Course Coordinator: Dr Tony M T Chan, Grad Dip, MPhil (CUHK), PhD (CityU)

Course Developer: The Open University, UK, Course Team

This course will be of particular interest to you if you need to implement any kind of strategy for decision making at your work. The course deals with how optimization problems arise in the business world, and the numerical skills and techniques that are needed to represent real optimization problems in terms of mathematical models and to interpret the solution from these models in relation to the decision making problem.

Many of the models discussed in the course are the most commonly used decision models that arise in industry, science, technology, business, economics and management. These include how to get the most revenue from mining china clay, solving an inventory problem and designing an optimized schedule, etc. Teaching is supported and enhanced by the use of the computer algebra package MathCAD.

MATH S373 is one of the higher-level courses for programmes in Mathematical Studies, and Statistics and Decision Science. This course is presented at 12-month intervals.

Advisory prerequisite(s)
Any student taking this course will need to have a firm understanding of mathematics. You are advised to have already studied one of the following mathematics courses: MATH S221, MATH S222 or MATH S215, or gained equivalent level before studying this course.

This course aims to:

• Introduce a broad area of optimization and operational research techniques that can be used for modelling real-life decision problems;
• Develop students' knowledge and understanding of the creation of mathematical models and their numerical solutions that arise in science, technology, business, economics and management;
• Enable students to formulate a real problem in mathematical terms, and to solve it numerically with computer software;
• Instruct students to make observations of the numerical quantities relevant to the solution of the decision making problems;
• Provide practical training in interpreting the solution in relation to the real problem and evaluating the success or failure of the mathematical model.

Contents
The course covers the following topics:

Block I

• Introduction to decision models
• Systems of linear system models and non-linear system models
• Introduction to iterative methods
• Ill-conditioning and induced instability
• Least squares method
• Mathematical modelling and applications:
  – the inventory problem.
  – aircraft flight optimization program.

Block II

• Linear programming
  – the simplex method and the two-phase simplex method
• Integer programming
  – the branch-and-bound method and models using 0–1 variables
• Applications of linear and integer programming
  – the machine scheduling problem
  – a mining transformation problem
  – the travelling salesman problem.

Block III

• Minimizing a function of one variable, two variables and n variables
  – interval reduction methods, alternating variables method, steepest descent method, Newton–Raphson method, and Gauss–Newton method
• Constrained non-linear optimization
  – quadratic penalty-function method, Newton–Lagrange method, augmented Lagrangian method, inequality-constraint models, and penalty-function methods
• Applications of optimization problem-solving
  – the factory location problem
  – the landmine problem
  – the grain chute problem
  – the oil pipeline problem.

Learning support
There will be 12 two-hour tutorials and six surgeries throughout the course.

Assessment
There are four assignments (from which the best three scores will be used to determine the final assignment grade) and a final examination. Students are required to submit assignments via the Online Learning Environment (OLE).

Online requirement
This course is supported by the Online Learning Environment (OLE). You can find the latest course information from the OLE. Through the OLE, you can communicate electronically with your tutor and the Course Coordinator as well as other students. To access the OLE, students will need to have access to the Internet. The use of the OLE is required for the study of this course and you can use it to submit assignments.

Equipment
Students will need access to a computer with a DVD drive in order to use the course software MathCAD and to watch video programmes provided with the course, and a scientific calculator.

To ensure the successful MathCAD installation, your computer needs to have the following minimum requirements:

• A computer with Windows XP or above
• Display resolution of 800 x 600
• Printer

Software
For this course students will use the software MathCAD 14 Professional, which is supplied as part of the course material. This software requires Microsoft Windows XP or higher. The teaching computing packages running in MathCAD will be supplied to students.

Set book(s)
There are no set books for this course.

Student with disabilities or special educational needs
The audio and visual components of this course may cause difficulties for students with an audio or visual impairment. You are encouraged to seek advice from the Course Coordinator before enrolling on the course.