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Show d2x dt2 dx dt + a2X + bu ( 7) with the initial conditions dt ~ n x( 0)=? o Define the following notation: dx Let x A ^~ xiAx ' 1& A dx x2= dt *. * to Then X] = ^ d2x X2 = d^ and the one second order differential equation can now be written as two first order differential equations: xi = x2 X2 = a] X2 + a2Xi+ bu with the initial conditions xi( 0)= 70 x2( 0)= 71 The latter two differential equations can be written in a compact matrix form as follows: _ x = Ax_ + Bu . x( 0)= 7 xi _ 0 r xi + 0 x2 a2 - & i x2 1 xi( 0) x2( 0) - - 7o " r where, A=; 5 1 a2al , and B = t- i o 1 _ - In summary, the overall mathematical model for the km subsystem k= 1,2,..., 6 ( in the first level- second layer) of the hierarchical structure can be written as follows: minimize f^ ( u_, x, a) u^ Subject to x_ = Ax_ + Bu_ x( 0)= 1 y = Cx_ + Du_ ( assuming linear input- output relationships) or X = M ( x » iL'£.) 0n a general case) and gj ( x, u_, a) < 0, j = 1, 2,..., Jfc The above functions will be constructed by integrating all the study's efforts. The proper construction of the functional relationships fj(*), g;(*), Hk(*) and x #,(•) will determine the worth of the overall management model developed in the study. The following section addresses the problem of decisionmaking when multiple objectives are in competition or conflict with one another. It is clear, however, that iO matter how good the procedures are for analyzing trade- offs among all the objectives, the quality of the overall model will depend on the realism and proper representation in the construction of these functions. The Surrogate Worth Trade- off Method The purpose of this section is to present a rather general and qualitative discussion on the SWT Method as it relates to the hierarchical structure of Figure 7. In addition, basic definitions related to the concept of Pare to optimum will be introduced- For a detailed discussion on the SWT method, the reader is referred to Haimes and Hall ( 1974), or to Haimes, Hall, and Freedman ( 1975). Assuming that there is one objective function associated with each of the six major goals and considerations ( first level- second layer of the hierarchical structure). The overall multiobjective higher level optimization problem can be formulated as follows: maximize f fj ( x, u, a), l^ ( x_, H. » £.) » •-• _ u_ V f6( x, u, a)} subject to the previously discussed system of constraints and input- output relationships. 47 |