11.6 DESIRED STRUCTURE OF THE OBJECTIVE FUNCTIONS TO BE OPTIMIZED USING THE TTT TRANSFORM AND THE GRAPHICAL OPTIMIZATION PROCEDURE

Past studies [1, 2, 3] demonstrate that every problem which can be transformed into a problem of maximizing or minimizing an expression of the form

(11.77)

or

(11.78)

can be analyzed by the TTT transform technique.

Now let's first look at the maximum or minimum point of Eq. (11.78), u*. Once u* is determined, the optimum point of Eq. (11.77), T*, can be obtained by solving

F( T*) = u*

for T*. Note that the maximization or minimization of Eq. (11.78) is equivalent to solving the following differential equation for u:

which leads to

or

(11.79)

Dividing both sides of Eq. (11.79) by ( ? ?) and rearranging yields

(11.80)

Note that the left side of Eq. (11.80), ?'( u*), is the slope of ?( u) at Point [ u*, ?( u*)], and the right side of Eq. (11.80),

is the slope of a straight line through Points [ u*, ?( u*)] and ( - ?/ ?, - ?/ ?). Therefore, Eq. (11.80) implies that the maximum or minimum value of C( u) occurs at a point C where the tangent line of ?( u) goes through Point ( - ?/ ?, - ?/ ?) as shown in...