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The series covers comprehensively all aspects of Operations Research. The scope of a Tutorial is to provide an understanding of an advanced topic to young researchers, such as PhD students or Post-docs, but also to senior researchers and practitioners. Tutorials may be used as textbooks in graduate courses.
These tutorials are not collections of chapters written by different authors. Mathematical Programming is the title given to a collection of optimisation algorithms that deal with constrained optimisation problems.
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Here the problems considered will all involve constraints which are linear, and for which the objective function to be maximised or minimised is also linear. Some of these problems are not continuously differentiable; special algorithms have to be developed.
The module considers first how these problems arise from practical applications, then introduces the solution of such problems, and finally explain the important area of post-optimality analysis where we answer questions about the effects of changes in the parameters of the problem on the optimal solution.
Most of OR business problems you will hear about come from the logistics industry. Here are some examples:.
When Operations Research was first invented, mathematicians recognized that most business problems they encountered could be mapped to a number of lower level generic families of problems. Hence, they dedicated a lot of their time and bandwidth to study these problems, gave them specific names, and proposed solutions for them.
Most of the OR practitioners I know and myself included spend most of their time on the process of converting a business problem to one of these well known generic problems. Once you get to that stage, solving the problem becomes standard procedure more or less.
Generic Operations Research problems are concise enough to be described in mathematical notation.
IFORS Tutorial Lecture
However, Operations Research practitioners usually express these problems using higher-level modelling languages. As an example of the higher-level modelling language, scheduling problems are often describe using terms like activities, resources and precedence constraints. We discussed how Generic Operations Research Problems can be expressed using higher level languages.
These notations are usually problem specific.
OR-Notes - contents
For instant, resources and activities can be used to describe a scheduling problem, but not TSP. You can however, describe any of these Generic OR problems or discreet optimization problems using a number of Modelling Paradigms.
A modelling paradigm is a set of rules and practices that allows you to represent higher level problems using lower-level data structures such as matrices. When using a modelling paradigm, the OR practitioner expresses the problem using mathematical notation or an Algebraic Modelling Language AML that converts these mathematical notations to matrices to pass to the last level of abstraction, algorithms. These are modelling paradigms that can express problems using linear equality constraints. It is a very powerful and natural way to represent optimization problem.
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Constraint Programming is another paradigm that has gained a lot of popularity lately, especially for scheduling applications. Network models are also very well known modelling paradigm based on graph theory. This paradigm is particularly useful to build good intuition of the problem and can represent a wide variety of discreet optimization problems in an efficient manner.
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An algorithm is a procedure or a sequence of steps that if followed can solve a problem. Some algorithms are ubiquitous in all fields of Computer Science like searching and sorting, while others are geared towards more specific problems. Search algorithms are very important in solving Operations Research problems.