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Show y_ = P- dimensional vector of outputs g; ( x, u, a) = the j" 1 constraint function; j= l, 2,..., J fjfa, u, a) = i" 1 objective function; ~ i= 1,2,..., I Although, both the objective functions and the constraints depend on the output vector,_ y_, the latter does not appear explicitly as an argument in fj (•) or gj (.)• This is due to the fact that the output vector, y_, is generally represented by a functional relationship of the form: ( where H( « ) is a P- dimensional vector of differentiate functions). Consequently,^ can be substituted in terms ofx, _ u, and a. in the functions fj (•) or gj( « ). The simplest functional relationship for H (.) is a linear one: j£ = Cx + Du_ where C and D are matrices of coefficients. The constraints g;(.) can be generalized to include all equality and inequality constraints. Therefore, the general form gj ( x, u_, a)< 0, j = 1,2,..., J will be used in this study. In this formulation, the exogeneous variables, a, will be assumed known. In general, they are determined either by other models, or by a parameter estimation and system identification process. To clarify the above mathematical notation a simple example follows: Consider the third objective in the hierarchical structure in Figure 7 to maximize oil production from the Great Salt Lake subject to all other environmental, societal, and other constraints and objectives. The following identification of variables in the oil drilling sub- model should not be considered all- inclusive but rather a selected sample for pedagogical purposes only: ( hi) piping, or $ investment ( iv) methods of exploration and production ( v) etc. a = 0) reservoir characteristics ( e. g. transmissivity, storativity, etc.) 00 price of crude oil ( iii) other values for operation maintenance and replacement ( OMR) cost or benefits ( iv) demand for oil ( v) etc. X. = 6) oil production ( ii) oil spill ( iii) employment opportunities ( iv) additional regional development due to this industry ( v) etc. h ( x , u_, a) = Net benefits from oil production * 1 & ±> a) = Oil spill < bj S2 fe , u_,_ a) = Other ecological effects < b2 Note that the relationship between the output vector, y_, and all other variables ( x_, u_, ja) through the functional relation, H, permits a state variable to be also a decision variable as well as an output. Consider for example the linear relation ( for the scalar case) y = cx + du for c• - 0, d =£ 0 then y = du, u = the production rate ( bbl/ day), d = number of production days ( days), y = total production in ( bbl). Often, the state of the system changes with time. For example, the pressure in the reservoir will drop with continuing production ( which in turn will result in a higher production cost), the water level in the Great Salt Lake will change, etc. These dynamic changes can be expressed in a system of differential equations. A first order linear system of differential equations can be witten as follows: x = u = 0) capacity of the crude oil natural reservoir CD pressure in the oil reservoir ( iii) depth of oil formation ( iv) water level in the Great Salt Lake ( v) etc. ( 0 production rate of crude oil ( ii) location and number of drilling facilities dx dt Ax + Bu With the initial conditions x_ ( 0) = x_ Q. Note that the vector x_, is a time variable vector, x_( t). Higher order differential equations also can be assumed. It is common in that case to introduce the so- called state space notation for compact model formulation. Given for example a second order differential equation: 46 |