#### Model Development

The quantitative analysis begins one the problem has been structure. Imagination, teamwork, and considerable effort are usually required to transform a rather general problem description into a well defined problem that can be approached via quantitative analysis.

When both the manager and the quantitative analyst agree that the problem has been adequately structured, work can begin on developing a model to represent the problem mathematically.

Solution procedures can then be employed to find the best solution for the model.

This best solution for the model then becomes a recommendation to the decision maker.

Models are representations of real objects or situations. In our case we will be interested in mathematical models. The purpose, or value of any model is that it enables us to make inferences about the real situation by studying and analyzing the model.

In general, experimenting with models requires less time and less expense than experimenting with real objects or situations.

The value of model-based conclusions and decisions is dependent on how well a model represents a real situation. The most closely a mathematical model represents the true relationships between variables, the more accurate the model outcomes will be.

When initially considering a managerial problem, we usually find that the problem structuring phase leads to a specific objective, such as maximization or minimization of the dependent variable subject to a set of restrictions or constraints.

The success of the mathematical model will depend heavily on how accurate the objective and constraints can be expressed in terms of mathematical equations or relationships.

The mathematical expression that describes the problem’s objective is referred to as the objective function.

Environmental factors, which can affect both the objective function and the constraints are referred to as uncontrollable inputs to the model. Controllable inputs are the decision variables.

Once all controllable and uncontrollable inputs are specified, the objective function and constraints can be evaluated, and the output of the model determined.

In this sense, the output of the model is simply the projection of what would happen if those particular environmental factors and decisions occur in the real situation.

As stated earlier, the uncontrollable inputs are those the decision makers cannot influence. The specific controllable and uncontrollable inputs of a model depend on the particular problem or decision-making situation.

If all uncontrollable inputs to a model are known and cannot vary, the model is referred to as a deterministic model.

The distinguished feature of a deterministic model is that the uncontrollable input values are known in advance.

If any of the controllable inputs are uncertain and subject to variation, the model is referred to as stochastic or probabilistic model.

An uncontrollable input to many production planning models is demand for the product.

In the production model, the number of hours of production time required per unit, the total hours available, and the unit profit all were uncontrollable inputs. Because the uncontrollable inputs were well known to take on fixed values, the model is deterministic.

Herbert A Simon, a Nobel prize winner in economics and an expert in decision making, said that a mathematical model does not have to be exact; it’s just has to be close enough to provide better results than can be obtained by common sense.