9.4 Computational Procedure in Dynamic Programming 553
the optimum value offi=
∑i
k= 1 Rkfor any specified value of the inputsi+^1. This
problem, by using the principle of optimality, has been decomposed intoi separate
problems, each involving only one decision variable. Equation (9.16) is the desired
recurrence relationship valid fori= 2 , 3 ,... , n.
9.4 Computational Procedure in Dynamic Programming
The use of the recurrence relationship derived in Section 9.3 in actual computations is
discussed in this section [9.10]. As stated, dynamic programming begins by subopti-
mizing the last component, numbered 1. This involves the determination of
f 1 ∗(s 2 ) =opt
x 1
[R 1 (x 1 , s 2 )] (9.17)
The best value of the decision variablex 1 , denoted asx∗ 1 , is that which makes the
return (or objective) functionR 1 assume its optimum value, denoted byf 1 ∗. Bothx 1 ∗
andf 1 ∗depend on the condition of the input or feed that the component1 receives from
the upstream, that is, ons 2. Since the particular values 2 will assume after the upstream
components are optimized is not known at this time, this last-stage suboptimization
problem is solved for a “range” of possible values ofs 2 and the results are entered
into a graph or a table. This graph or table contains a complete summary of the results
of suboptimization of stage 1. In some cases, it may be possible to expressf 1 ∗as a
function ofs 2. If the calculations are to be performed on a computer, the res ults of
suboptimization have to be stored in the form of a table in the computer. Figure 9.8
shows a typical table in which the results obtained from the suboptimization of
stage 1 are entered.
Next we move up the serial system to include the last two components. In this
two-stage suboptimization, we have to determine
f 2 ∗(s 3 ) =opt
x 2 ,x 1
[R 2 (x 2 , s 3 )+R 1 (x 1 , s 2 )] (9.18)
Since all the information about component 1 has already been encoded in the table
corresponding tof 1 ∗, this information can then be substituted forR 1 in Eq. (9.18) to
Figure 9.8 Suboptimization of component 1 for various settings of the input state variables 2.
