High School Statutory Authority: Algebra I, Adopted One Credit.

In deterministic models good decisions bring about good outcomes. You get that what you expect; therefore, the outcome is deterministic i. This depends largely on how influential the uncontrollable factors are in determining the outcome of a decision, and how much information the decision-maker has in predicting these factors.

Those who manage and control systems of men and equipment face the continuing problem of improving e. The problem may be one of reducing the cost of operation while maintaining an acceptable level of service, and profit of current operations, or providing a higher level of service without increasing cost, maintaining a profitable operation while meeting imposed government regulations, or "improving" one aspect of product quality without reducing quality in another.

To identify methods for improvement of system operation, one must construct a synthetic representation or model of the physical system, which could be used to describe the effect of a variety of proposed solutions. A model is a representation of the reality that captures "the essence" of reality.

A photograph is a model of the reality portrayed in the picture. Blood pressure may be used as a model of the health of an individual. A pilot sales campaign may be used to model the response of individuals to a new product. In each case the model captures some aspect of the reality it attempts to represent.

Since a model only captures certain aspects of reality, it may be inappropriate for use in a particular application for it may capture the wrong elements of the reality.

Temperature is a model of climatic conditions, but may be inappropriate if one is interested in barometric pressure.

A photograph of a person is a model of that individual, but provides little information regarding his or her academic achievement. An equation that predicts annual sales of a particular product is a model of that product, but is of little value if we are interested in the cost of production per unit.

Thus, the usefulness of the model is dependent upon the aspect of reality it represents. If a model does capture the appropriate elements of reality, but capture the elements in a distorted or biased manner, then it still may not be useful.

An equation predicting monthly sales volume may be exactly what the sales manager is looking for, but could lead to serious losses if it consistently yields high estimates of sales. A thermometer that reads too high or too low would be of little use in medical diagnosis.

A useful model is one that captures the proper elements of reality with acceptable accuracy. Such problems arise in all areas of business, physical, chemical and biological sciences, engineering, architecture, economics, and management.

The range of techniques available to solve them is nearly as wide. A mathematical optimization model consists of an objective function and a set of constraints expressed in the form of a system of equations or inequalities. Optimization models are used extensively in almost all areas of decision-making such as engineering design, and financial portfolio selection.

This site presents a focused and structured process for optimization analysis, design of optimal strategy, and controlled process that includes validation, verification, and post-solution activities. If the mathematical model is a valid representation of the performance of the system, as shown by applying the appropriate analytical techniques, then the solution obtained from the model should also be the solution to the system problem.

The effectiveness of the results of the application of any optimization technique, is largely a function of the degree to which the model represents the system studied. To define those conditions that will lead to the solution of a systems problem, the analyst must first identify a criterion by which the performance of the system may be measured.

This criterion is often referred to as the measure of the system performance or the measure of effectiveness.

In business applications, the measure of effectiveness is often either cost or profit, while government applications more often in terms of a benefit-to-cost ratio. If the objective function is to describe the behavior of the measure of effectiveness, it must capture the relationship between that measure and those variables that cause it to change.

System variables can be categorized as decision variables and parameters. A decision variable is a variable, that can be directly controlled by the decision-maker. There are also some parameters whose values might be uncertain for the decision-maker.

This calls for sensitivity analysis after finding the best strategy. In practice, mathematical equations rarely capture the precise relationship between all system variables and the measure of effectiveness. This mathematical relationship is the objective function that is used to evaluate the performance of the system being studied.

Formulation of a meaningful objective function is usually a tedious and frustrating task. Attempts to develop the objective function may fail. Failure could result because the analyst chose the wrong set of variables for inclusion in the model, because he fails to identify the proper relationship between these variables and the measure of effectiveness.

Returning to the drawing board, the analyst attempts to discover additional variables that may improve his model while discarding those which seem to have little or no bearing. However, whether or not these factors do in fact improve the model, can only be determined after formulation and testing of new models that include the additional variables.

The entire process of variable selection, rejection, and model formulation may require multiple reiteration before a satisfactory objective function is developed.

The analyst hopes to achieve some improvement in the model at each iteration, although it is not usually the case. Ultimate success is more often preceded by a string of failures and small successes.Deterministic modeling process is presented in the context of linear programs (LP).

LP models are easy to solve computationally and have a wide range of applications in diverse fields. This site provides solution algorithms and the needed sensitivity analysis since the solution to a practical problem is not complete with the mere determination of the optimal solution.

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1oa1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

1oa2 Solve word problems that call for addition of three whole numbers whose sum is less. Deterministic modeling process is presented in the context of linear programs (LP). LP models are easy to solve computationally and have a wide range of applications in diverse fields.

This site provides solution algorithms and the needed sensitivity analysis since the solution to a practical problem is not complete with the mere determination of the optimal solution. A Diophantine equation is a polynomial equation whose solutions are restricted to integers.

These types of equations are named after the ancient Greek mathematician Diophantus. A linear Diophantine equation is a first-degree equation of this type. Diophantine equations are important when a problem requires a solution in whole amounts.

The study of problems that require integer solutions is. Pearson Prentice Hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum.

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Linear Optimization