1. Computational Methods for Management and Economics Carla Gomes Module 3 OR Modeling Approach

2. Overview of OR Modeling Approach

3. OR Nature Operations research involves “research” –approach resembles the scientific method OR – “search for optimality” (best, optimal solution) is an important theme in OR. OR - team approach (mathematics, statistics and probability theory, economics, business administration, computer science, and engineering and other areas relevant to the particular organization OR adopts an organizational point of view – i.e., it tries to meet the goals of the overall organization.

4. Overview of OR Modeling Approach 1. Define the problem of interest and gather the relevant data 2.Formulate a mathematical model to represent the problem 3.Develop a computer base procedure for deriving solutions to the problems from the model 4.Test the model and refine it as needed 5.Prepare for ongoing application of the model as prescribed by the management 6.Implement

5. 1. Define the problem of interest and gather the relevant data Need to develop a well-defined statement of the problem to be considered. objectives, constraints (e.g., what can be done with available resources) inter-relationships between the area to be studied and other areas in the organization possible alternative course of actions time limits for making a decision... Difficult phase - ill-defined nature and it’s difficult to be taught. It depends on the particular problem and domain of activity.

6. 2. Formulate a mathematical model to represent the problem Formulate the problem into a way that is convenient for analysis – typically using a mathematical model. Mathematical models – idealized representations expressed in terms of mathematical symbols and expressions. Famous mathematical models F = ma (Newton’s second law of motion) and E=mc² (Einstein’s famous equation of conservation of energy into mass).

7. Mathematical model of a business problem: Decision variables - they represent quantifiable decisions to be made, under our control, say x 1, x 2, …, x n. The respective values are to be determined. Objective function – expresses the appropriate measure of performance as a mathematical function of the decision variables. E.g., P = 2 x 1 + x 2 + …+ x n Constraints – any restriction on the values of the variables are also expressed mathematically, typically by means of equations (e.g. 2 x 1 + 3 x 2 <= 10) Parameters of the model – constants (namely coefficients and right- end-sides) in the constraints and objective function. Typical OR model Choose the decision variables so as to maximize (or minimize) the objective function, subject to the specified constraints.

8. Mathematical Program Optimization problem in which the objective and constraints are given as mathematical functions and functional relationships. Optimize: Z = f(x 1, x 2, …, x n ) Subject to: g 1 (x 1, x 2, …, x n ) =,, b 1 … g 2 (x 1, x 2, …, x n ) =,, b 2 g m (x 1, x 2, …, x n ) =,, b m

9. Comments: a) Gathering of the relevant data --- frequently difficult.  value assigned to parameters are often rough estimates.  it is important to analyze how the solution derived from the model would change if the value assigned to the parameters (one at a time) were changed to other plausible values. This process is referred to as sensitivity analysis (discussed later).

10. b) In general real problems can be modeled using more than one model.  The process of testing a model typically leads to a succession of models that provide better and better representations of the problem.  Even possible that two or more different LP models completely different types of models may be developed for the same problem  Even possible that two or more completely different types of models may be developed to help analyze the same problem.

11. 3. Develop a computer base procedure for deriving solutions to the problems from the model  This phase can be very easy if we are using a well known mathematical model such as LP.

12. 4. Test the model and refine it as needed – model validation Developing a mathematical model is like developing a large computer program – in general it has bugs! Some tips : Fresh look at the model to check for obvious errors or oversights (including a new person who didn’t participate in the original process) Retrospective test – use of historical data to reconstruct the past --- how well the model and the resulting solution would have performed if they had been used. (even though the key issue about using the model is to predict the future and things may change…) Careful technical review of the model by individuals not involved in the design of the model

13. 5 Prepare for ongoing application of the model as prescribed by the management Install a well documented system for applying the model. The system includes: Mathematical Model Solution procedure (including post-optimality procedure) Operation procedures for implementation In general --- computer-based system often integrating databases and management information systems. Quite often interactive system (Decision Support Systems) are used.

14. 6. Implementation Critical phase to make sure that the model and recommendations of precious phases are properly implemented. The benefits of the study are reaped only after this phase.

15. Key Aspects: Management support Involvement of users of system - so that they feel also ownership of the system and they don’t reject the system Feedback on how well the system is working and if the assumptions of the system continue to be satisfied throughout the entire period during which the system is used. Gradually phase in the system while identifying and eliminating flaws. Adoption of management incentives for the effective implementation of the system.

16. Advantages of mathematical models : describe a problem in a very concise way provide a bridge to sue very powerful computer packages Make sure that the model is a valid representation of the problem – high correlation between the prediction by the model and what would actually happen in the real world(much of the model validation work is performed during the testing phase). We should be able to solve the model – “tractability”. Pitfalls to avoid when using mathematical models: