In general, experimenting with models takes less time and is less expensive than experimenting with the real object or situation. A model aircraft is certainly faster and less expensive to build and study than the full-scale aircraft. Likewise, the mathematical model in equation (1.1) allows for quick identification of profit expectations without actually requiring the manager to produce and sell x units. Models also have the advantage of reducing the risk associated with experiencing the real situation. In particular, bad designs or bad decisions that cause a model airplane to crash or a mathematical model to project a loss of $10,000 can be avoided in the real situation. The value of conclusions and decisions based on the model depends on how well the model represents the situation. real situation. The more closely the aircraft model represents the real aircraft, the more accurate the conclusions and predictions will be. Likewise, the more closely the mathematical model represents the company's true profit-volume ratio, the more accurate the profit projections will be. Since this text deals with quantitative analyzes based on mathematical models, let's take a closer look at the mathematical modeling process. When we initially consider a management problem, we usually find that the problem definition phase leads to a specific objective, such as profit maximization or cost minimization, and possibly a set of restrictions or constraints, such as production capacities. The success of the mathematical model and quantitative approach will largely depend on how precisely the objective and constraints can be expressed in terms of mathematical equations or relationships. A mathematical expression that describes the problem or...... middle of paper... ... The time required to prepare this data and the possibility of errors in data collection will make the data preparation phase a critical part of the quantitative analysis process. Often a large enough database is needed to support a mathematical model, and information systems specialists may be involved in the data preparation phase. Model Solution Once the model development and data preparation phases are complete, we can proceed to the model solution phase. In this step, the analyst will attempt to identify the values ​​of the decision variables that provide the “best” output for the model. The specific value or values ​​of the decision variable that provide the “best” result will be defined as the optimal solution for the model. For the production problem, the model solution step involves finding the value of the production quantity decision variable x that maximizes profit