In using a mathematical model the idea is to first capture all the crucial aspects of the system using the three element In using a mathematical model the idea is to first capture all the crucial aspects of the system using the three elements just described, and to then optimize the objective function by choosing from among all values for the decision variables that do not violate any of the constraints specified the specific values that also yield the most desirable maximum or minimum value for the objective function. This process is often called mathematical programming. Although many mathematical models tend to follow this form, it is certainly not a requirement; for example, a model may be constructed to simply define relationships between several variables and the decisionmaker may use these to study how one or more variables are affected by changes in the values of others. Decision trees, Markov chains and many queuing models could fall into this category. Before concluding this section on model formulation, we return to our hypothetical example and translate the statements made in the problem definition stage into a mathematical model by using the information collected in the data collection phase. To do this we define two decision variables G and W to represent respectively the number of gizmos and widgets to be made and sold next month.Fassilia United States. Alabama
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Very often, one may simultaneously have more than one objective function to optimize e. Very often, one may simultaneously have more than one objective function to optimize e. , maximize profits and minimize changes in workforce levels, say . In such cases there are two options. First, one could focus on a single objective and relegate the others to a secondary status by moving them to the set of constraints and specifying some minimum or maximum desirable value for them. This tends to be the simpler option and the one most commonly adopted. The other option is to use a technique designed specifically for multiple objectives such as goal programming .Fassilia United States. Alabama
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Clearly, constraints dictate the values that can be feasibly assigned to the decision variables, i. Clearly, constraints dictate the values that can be feasibly assigned to the decision variables, i. , the specific decisions on the system or process that can be taken. The third and final component of a mathematical model is the objective function. This is a mathematical statement of some measure of performance such as cost, profit, time, revenue, utilization, etc. and is expressed as a function of the decision variables for the model. It is usually desired either to maximize or to minimize the value of the objective function, depending on what it represents.Fassilia United States. Alabama
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An analysis of the model will seek specific values for these variables that are desirable from one or more perspectives. An analysis of the model will seek specific values for these variables that are desirable from one or more perspectives. Very often especially in large models it is also common to define additional convenience variables for the purpose of simplifying the model or for making it clearer. Strictly speaking, such variables are not under the control of the decisionmaker, but they are also referred to as decision variables. Constraints are used to set limits on the range of values that each decision variable can take on, and each constraint is typically a translation of some specific restriction e. , the availability of some resource or requirement e. , the need to meet contracted demand .Fassilia United States. Alabama
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As an illustration, the Critical Path Method CPM and the Program Evaluation and Review Technique PERT are two very simil As an illustration, the Critical Path Method CPM and the Program Evaluation and Review Technique PERT are two very similar O. techniques used in the area of project planning. However, CPM is based on a deterministic mathematical model that assumes that the duration of each project activity is a known constant, while PERT is based on a probabilistic model that assumes that each activity duration is random but follows some specific probability distribution typically, the Beta distribution . Very broadly speaking, deterministic models tend to be somewhat easier to analyze than probabilistic ones; however, this is not universally true. Most mathematical models tend to be characterized by three main elements decision variables, constraints and objective function s . Decision variables are used to model specific actions that are under the control of the decisionmaker.Fassilia United States. Alabama
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