Engineering Optimization: Theory and Practice, Fourth Edition

9.3 Concept of Suboptimization and Principle of Optimality 551

Figure 9.7 Suboptimization (principle of optimality).

suboptimization is shown in Fig. 9.7. Since the suboptimizations are to be done in the
reverse order, the components of the system are also numbered in the same manner for
convenience (see Fig. 9.3).
The process of suboptimization was stated by Bellman [9.2] as the principle of
optimality:

An optimal policy (or a set of decisions) has the property that whatever the

initial state and initial decision are, the remaining decisions must constitute

an optimal policy with regard to the state resulting from the first decision.

Recurrence Relationship. Suppose that the desired objective is to minimize the
n-stage objective functionf, which is given by the sum of the individual stage returns:

Minimizef=Rn(xn, sn+ 1 )+Rn− 1 (xn− 1 , sn) +· · · +R 1 (x 1 , s 2 ) (9.9)

where the state and decision variables are related as

si=ti(si+ 1 , xi), i= 1 , 2 ,... , n (9.10)