Formulation of the Objective Function

Formulation of the Objective Function The formulation of objective functions is one of die crucial steps in the apphcation of optimization to a practical problem. You must be able to translate the desired objective to mathematical terms. In the chemical process industries, the objective function often is e ressed in units of currency per unit time (o-g-, U.S. dollars per week, month, or year) because the normal industrial goal is to minimize costs or maximize profits subject to a variety of constraints. 

A typical economic model involves the costs of raw materials, values of products, and costs of production as functions of operating conditions, projected sales figures, and the like. An objective function can be e ressed in terms of these quantities e.g., annual operating profit ( yyr) might be expressed as 

Iw = sum of product flow rates times respective product values (income) 

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Over the years two ML estimation approaches have evolved (a) parameter estimation based an implicit formulation of the objective function and (b) parameter and state estimation or "error in variables" method based on an explicit formulation of the objective function. In the first approach only the parameters are estimated whereas in the second the true values of the state variables as well as the values of the parameters are estimated. In this section, we are concerned with the latter approach. 

Direct measured variables are x, y, P, and T, while y may be seen as an indirect measurable quantity. Different formulations of the objective function have been studied. The following two alternatives are recommended ... 

Release time of units, i.e. the time needed for any unit to become available, is incorporated in the formulation of the objective function... 

In the formulation of the objective function there are two conflicting elements to produce as much valuable product as possible, but using as little energy as possible. For a given feed, the cost function is defined as the amount of propane, butane and pentane from the LP column (at 0.99 mol% or more) multiplied by the relevant product prices, minus the cost of boilup ... 

The Index I Is related to a specific design situation. The essential disadvantage of the formulation of eq.(2) Is Its restriction to one particular partial damage state, Which In fact may be sufficient for some structural systems but for general application several partial damage states have to be considered. This, however, requires a more general formulation of the objective function, I.e. the Incorporation of several partial damage states. 

The discussion of the suggested method for reliability based optimization Is performed in terms of comparative calculations. For this purpose earlier formulations of the objective function are applied. Furthermore the system reliability analysis Is confined to first order analysis Instead of the more sophlstticated response surface approach. For a complete comparison local as well as global damage has to be taken into account. Therefore three formulations were chosen ... 

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