Page "learned" Paragraph 1372

To see how important this economy is, let us suppose that there are M operating variables at each stage and that the state is specified by N variables ; ;

then the search for the maximum at any one stage will require a number of operations of order Af ( where a is some number not unreasonably large ).

To proceed from one stage to the next a sufficient number of feed states must be investigated to allow for interpolation ; ;

this number will be of the order of Af.

If, however, we are seeking the optimal R-stage policy for a given feed state, only one search for a maximum is required at the final step.

Thus a number of operations of the order of Af are required.

If all the operating variables were varied simultaneously, Af operations would be required to do the same job, and as R increases this increases very much more rapidly than the number of operations required by the dynamic program.

But even more important than this is the fact that the direct search by simultaneously varying all operating conditions has produced only one optimal policy, namely, that for the given feed state and R stages.

In contrast, the dynamic program produces this policy and a whole family of policies for any smaller number of stages.

If the problem is enlarged to require a complete coverage of feed states, Af operations are needed by the dynamic program and Af by the direct search.

But Af is vastly larger than R.

No optimism is more baseless than that which believes that the high speed of modern digital computers allows for use of the crudest of methods in searching out a result.

Suppose that Af, and that the average operation requires only Af sec..

Then the dynamic program would require about a minute whereas the direct search would take more than three millennia!!