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# Department of Mathematical Sciences

# MATH3141 Operations Research III

As its name implies, operations research involves "research on operations", and it is applied to problems that concern how to conduct and coordinate the operations (activities) within an organization. The nature of the organization is essentially immaterial, and, in fact, OR has been applied extensively in such diverse areas as manufacturing, transportation, construction, telecommunications, financial planning, health care, the military, and public services to name just a few.

This course is an introduction to mathematical methods in operations research. Usually, a mathematical model of a practical situation of interest is developed, and analysis of the model is aimed at gaining more insight into the real world. Many problems that occur ask for optimisation of a function under some constraints. If the function and the constraints are all linear, the simplex method is a powerful tool for optimisation. This method is introduced and applied to several problems, e.g. within transportation.

Many situations of interest in OR involve processes with random aspects and we will introduce stochastic processes to model such situations. An interesting area of application, addressed in this course, is inventory theory, where both deterministic and stochastic models will be studied and applied. Further topics will be chosen from: Markov decision processes; integer programming; nonlinear programming; dynamic programming.

## Outline of Course

Aim: To introduce some of the central mathematical models and methods of operations research.

### Term 1

• Introduction to Operations Research: Role of mathematical models, deterministic and stochastic OR.
• Linear Programming: LP model; convexity and optimality of extreme points; simplex method; duality and sensitivity; special types of LP problems, e.g. transportation problem.
• Networks: Analysis of networks, e.g. shortest-path problem, minimum spanning tree problem, maximum flow problem; applications to project planning and control.

### Term 2

• Introduction to Stochastic Processes: Stochastic process; discrete-time Markov chains.
• Markov Decision Processes: Markovian decision models; the optimality equation; linear programming and optimal policies; policy-improvement algorithms; criterion of discounted costs; applications, e.g. inventory model.
• Inventory Theory: Components of inventory models; deterministic models; stochastic models.
• Further topics chosen from:
• Dynamic programming: Characteristics; deterministic and probabilistic dynamic programming.
• Queueing theory: Models; waiting and service time distributions; steady-state systems; priority queues.
• Integer programming: Model; alternative formulations; branch-and-bound technique; applications, e.g. knapsack problem, travelling salesman problem.
• Nonlinear programming: Unconstrained optimisation; constrained optimisation (Karush-Kuhn-Tucker conditions); study of algorithms; approximations of nonlinear problems by linear problems; applications.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.