Modelling Approach or Overview - Elements of Operation research, Business Mathematics and Statistics | Business Mathematics and Statistics - B Com PDF Download

OVERVIEW OF THE OR MODELLING APPROACH

OR is applied to problems that concern how to conduct and coordinate opera- tions(i.e. activities) within an organization. The major phases of a typical OR

study are the following:

1. Define the problem and gather relevant data.

2. Formulate a mathematical model to represent the problem.

3. Develop a computer-based procedure for deriving a solution to the problem resulting from the model.

4. Test the model and refine it as needed.

5. Prepare for the ongoing application of the model as prescribed by management.

6. Implement.

Now we will discuss these phases in turn starting with the process of defining the problem and gathering data which includes determining such things as:

• the appropriate objectives,

• constraints on what can be done,

• interrelations between the area to be studied and the other areas of organization,

• possible alternative courses of action

• time limits for making a decision

The second phase is to formulate a mathematical model. A mathematical model is defined by a system of equations and related mathematical expressions that describe

the essence of a problem. A main elements of a model are the following:

Decision variables If there are n related quantifiable decisions to be made, they are represented as decision variables x1,x2,...,xn whose values are to be determined.

Objective function The appropriate (overall) measure of performance(e.g. profit) is expressed as a mathematical function of the decision variables (for example

P = 5x1 + 2x2 + ˇˇˇ + 20xn).

Constraints Any restrictions on the values that can be assigned to the decision variables are expressed mathematically, typically by means of inequalities or equations (for example (x1)2 − 2x1x2 + (x2)2 ≤ 25).

Parameters of the model The constants (namely, the coefficients and right- hand sides ) in the constraints and the objective function.

The next phase is to solve the resulting model. The problem is to choose the values of the decision variables so as to optimize (maximize or minimize) the objective function, subject to the specific constraints. An important part of the model-building process is determining the appropriate values to assign to the parameters of the model. This requires gathering relevant data. Another important part of model solution is so called Sensitivity analysis. This aims to answer the following question: How would the solution derived from the model change (if at all) if the values assigned to the parameters were changed to other plausible values (There is uncertainty about true values of the parameters).

Deriving the solution from the model

• Develop a computer-based procedure for deriving the solution to the problem from the model-sometimes it possible to use standard OR algorithms (there

are a number of readily available software pacakages).

• Search for an optimal, or best, solution. These solutions are optimal only with respect to the model being used (since the model is an idealized, rather

then an exact, representation of the real problem).

• Use heuristic procedures (i.e. intuitively designed procedures that do not guarantee an optimal solution) to find a good suboptimal solution in case

where the time or cost required to find an optimal solution to the model is very large.

• Postoptimality analysis or What-if analysis. This answers the question what would happen to the optimal solution if different assumptions are made about future conditions? It also involves conducting sensitivity analysis to determine which parameters of the model are most critical in determining the solution.

• Test the model and refine it as needed.

In this textbook, we concentrate on formulating mathematical models commonly used in modelling decision makers’ problems and solving the mathematical models.

The main subjects of the textbook are following:

• Linear Programming

– Introduction to Linear Programming

– Solving Linear Programming Problems - The Simplex Method

– Duality Theory and Sensitivity Analysis

• Integer Programming

• Decision Analysis

• Multiple Objective Linear Programming and Goal Programming