OPERATIONS RESEARCH BASIC COURSE | MIO310 |

**Aim**

The course has the overarching theme of optimisation and simulation and aims to provide basic knowledge in deterministic and stochastic modelling of operational and managerial business problems.

*Knowledge and understanding*

For a passing grade the student must

- be able to use basic optimisation/math programming theory, queuing theory and methodology for discrete event simulation modelling, to analyse, and solve business problems relating to operational and managerial decisions.

- to be able to formulate linear and integer programming models and solve these using a commercial software (Excel).
- to be able to interpret solutions, results and sensitivity analysis obtained as output from the commercial software mentioned above.
- to be able to explain the theory behind the simplex method and to be able to use this method for solving linear programming problems (LP-problems).
- to be able to explain and use well known methods for sensitivity analysis of LP-problems.
- to be able to explain and use duality theory to analyse and solve LP-problems.
- to be able to explain and use basic branch-and-bound techniques for solving mixed integer programming problems (MIP).
- to be able to formulate optimisation/math programming problems (LP and MIP) with several objective functions according to the techniques of goal programming.

- to understand the principles of discrete event simulation modelling, and the opportunities and limitations this technique offers.
- to be able to use a commercial software (Extend) to create a simulation model and use this to analyse discrete event systems and processes.
- to be able to correctly use statistical methods to analyse input to, and output from simulation models, and to interpret the generated results. This involves the choice and fitting of distribution functions, as well as using various types of hypothesis testing methods.

- to be able to formulate relevant business problems characterised by uncertainty in the availability and demand for capacity as queuing problems.
- to understand and explain principles for analytical modelling of basic queuing systems. The focus is on analysis of systems with exponentially distributed inter-arrival times and service times. However, it is also expected from the students to be able to analyse certain aspects of systems with generally distributed inter-arrival and service times.
- to be able to compute steady-state probability distributions of events as well as mean values of various performance measures as waiting times, queue lengths and costs.
- to be able to interpret the solutions and results obtained from the queuing models and place them in a business context.

*Skills and abilities*

For a passing grade the student must

- optimisation/math programming models (LP, MIP, goal programming)
- queuing models (birth and death processes, M/M/c, M/M/c/K, M/M/c/¥/N, M/G/1)
- discrete event simulation models (modelled in the software Extend)

**Contents**

In the optimisation/math programming section of the course, the main concern is methods for linear and integer programming. The focus is on formulation, and interpretation of the results obtained when the formulated problems are solved using a commercial software. The purpose of using mathematical models to analyse decision problems is to provide the decision maker with a solid foundation for his actions. However, in order to use information produced by the models in a correct way requires an understanding for the underlying mathematics. Consequently, a significant part of the course is devoted to clarify basic mathematical methods used within the optimisation/mathematical programming area. The sections mandatory project assignment is based on a case study describing a relatively complex business problem where an optimal decision is sought. By formulating and analysing an LP-model of the problem situation, a detailed decision proposal should be presented in a well structured technical report. Important assignment activities are: formulation of a relevant model, evaluation and optimisation of the model using a commercial software, and interpretation and sensitivity analysis of the obtained results.

The simulation section of the course examines basic queuing theory as an analytical tool for analysing stochastic systems with relatively simple structure. To deal with more complex systems, a commercial software for discrete event simulation (Extend) is used. The developed models are used for analysing and improving production processes, and flow of materials and information. In order to arrive at a relevant simulation model, various types of stochastic events and processes must be characterised by appropriate distribution functions. Moreover, the output data from the simulation model must be analysed statistically in a correct way. Another important aspect is how to verify and validate the model to assure it is relevant and the results can be trusted. The sections mandatory project assignment is structured around a case study dealing with the analysis of a small production system using simulation models. The objective is to provide an understanding for the strengths and weaknesses with discrete event simulation models as a tool for process analysis. Each project group reports their assignment work and the obtained results in a well structured technical report.

The connections and integration of the optimisation and simulation sections of the course are illustrated through a project assignment. This assignment is structured around a described business case, which should be analysed using both linear programming and simulation. An important objective is to bring forward, the strengths and weaknesses with the different approaches, and the value of using them in an integrated fashion to analyse a typical production planning problem. Also in this assignment, each project group is required to document their work and their conclusions in a well structured technical report.

**Literature**

Hillier F. S. and G. J. Lieberman, Introduction to Operations Research, 8th edition, McGraw-Hill, 2005. Customised edition for MIO310.

Laguna M. and J. Marklund, Business Process Modeling, Simulation and Design, Prentice Hall, 2005. (Includes a CD with the software Extend and other complementary material)

Course compendium