Undergraduate Course: Linear Programming, Modelling and Solution (MATH10073)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 10 (Year 3 Undergraduate)
||Availability||Available to all students
|Summary||Linear programming (LP) is the fundamental modelling technique in optimal decision-making. This course will introduce the concepts of LP modelling, explore the mathematical properties of general LP problems and study the theory of the simplex algorithm as a solution technique. Students will use the Xpress mathematical programming system to create, solve and analyse case studies and then present their work in oral and written form. As a consequence, in addition to the assessment of theoretical understanding and hand calculation via a written examination, the course is also assessed via a short individual Xpress assignment and a substantial group-based case study.
Linear programming (LP) offers the natural entry to the study of operational research, not only because LP is the fundamental modelling technique in optimal decision-making, but also because the mathematical nature of LP problems [everything is linear!] means that they can be analysed with tools from linear algebra introduced at level 8. This course introduces the concepts of LP modelling, explores the mathematical properties of general LP problems and studies the theory of the simplex algorithm as a solution technique. The novel feature of this course is that it introduces the Xpress mathematical programming system to create, solve and analyse case studies. The course ends with a group-based case study in which, much like an OR consultant might do, you will model, solve and analyse a meaningful example, presenting your work in oral and written form.
1. Linear programming: Decision variables, objective function, bounds and constraints. The feasible region; geometric and algebraic characterisation of an optimal solution. The dual of an LP problem and duality theory. Theory underlying sensitivity and fair prices.
2. Modelling: Introduction to the Xpress mathematical programming system as a means of modelling, solving and analysing LP case studies. Exploration of the modelling language Mosel to define index sets, data arrays, decision variables, constraints, solve LP problems, analyse problem sensitivity and report the results in a suitable format for further processing using Excel.
3. Solution: Study of the simplex algorithm for LP problems. Geometric and algebraic concepts underlying the algorithm and consequences for solution methods. Proof of termination for non-degenerate LPs. Linear algebra underlying its implementation via the revised simplex method.
Information for Visiting Students
|Pre-requisites||Previous study of linear algebra: matrix (non-)singularity, linear systems of equations, matrix-matrix multiplication. Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling.
|High Demand Course?
Course Delivery Information
|Academic year 2022/23, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 15,
Seminar/Tutorial Hours 5,
Supervised Practical/Workshop/Studio Hours 10,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 50%, Examination 50%
||Hours & Minutes
|Main Exam Diet S2 (April/May)||MATH10073 Linear Programming, Modelling and Solution||2:00|
On completion of this course, the student will be able to:
- Model, solve and analyse a simple case study using Xpress and present an investigation of that case study in oral and written form.
- Use the simplex algorithm by hand to solve a small linear programming problem
- Use the optimality conditions for a linear programming problem to deduce properties of its optimal solution
- Form the dual of a (primal) linear programming problem and exploit the relations between the two problems
- Formulate and analyse linear programming models for solving specific classes of problem.
|Introduction to Operations Research, F. S. Hillier and G. Lieberman, McGraw-Hill Higher Education, 9th edition. ISBN-10: 0071267670|
|Graduate Attributes and Skills
||Experience of modelling realistic case studies. Further development of programming skills (using Mosel), group-work, verbal and oral presentation skills.
|Additional Class Delivery Information
||16 one-hour lectures
5 one-hour workshops
4 two-hour labs
|Keywords||LPMS,linear programming,modelling language,case study
|Course organiser||Dr Julian Hall
Tel: (0131 6)50 5075
|Course secretary||Miss Greta Mazelyte