The set of all decisions is called the operating policy or , more simply , the policy .
An optimal policy is one which in some sense gets the best out of the process as a whole by maximizing the value of the product .
There are thus three components to an optimal design problem : ( 1 )
The specification of the state of the process stream ; ;
( 2 )
The specification of the operating variables and the transformation they effect ; ;
( 3 )
The specification of the objective function of which the optimization is desired .
For a chemical process the first of these might involve the concentrations of the different chemical species , and the temperature or pressure of the stream .
For the second we might have to choose the volume of reactor or amount of cooling to be supplied ; ;
the way in which the transformation of state depends on the operating variables for the main types of reactors is discussed in the next chapter .
The objective function is some measure of the increase in value of the stream by processing ; ;
it is the subject of Chapter 4 .
The essential characteristic of an optimal policy when the state of the stream is transformed in a sequence of stages with no feedback was first isolated by Bellman .
He recognized that whatever transformation may be effected in the first stage of an R-stage process , the remaining stages must use an optimal Af-stage policy with respect to the state resulting from the first stage , if there is to be any chance of optimizing the complete process .
Moreover , by systematically varying the operating conditions in the first stage and always using the optimal Af-stage policy for the remaining stages , we shall eventually find the optimal policy for all R stages .
Proceeding in this way , from one to two and from two to three stages , we may gradually build up the policy for any number .
At each step of the calculation the operating variables of only one stage need be varied .
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 ! !
The principle of optimality thus brings a vital organization into the search for the optimal policy of a multistage decision process .
Bellman ( 1957 ) has annunciated in the following terms :
`` An optimal policy has the property that whatever the initial state and initial decision are , the remaining decisions must constitute an optimal policy with respect to the state resulting from the first decision '' .
This is the principle which we will invoke in every case to set up a functional equation .
It appears in a form that is admirably suited to the powers of the digital computer .
At the same time , every device that can be employed to reduce the number of variables is of the greatest value , and it is one of the attractive features of dynamic programming that room is left for ingenuity in using the special features of the problem to this end .
2.2 the discrete deterministic process
Consider the process illustrated in Fig. 2.1 , consisting of R distinct stages .
These will be numbered in the direction opposite to the flow of the process stream , so that stage R is the T stage from the end .
Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af .
Thus Af denotes the state of the feed to the R-stage process , and Af the state of the product from the last stage .
Each stage transforms the state Af of its feed to the state Af in a way that depends on the operating variables Af .
We write this Af .
This transformation is uniquely determined by Af and we therefore speak of the process as deterministic .
In practical situations there will be restrictions on the admissible operating conditions , and we regard the vectors as belonging to a fixed and bounded set S .
The set of vectors Af constitutes the operating policy or , more briefly , the policy , and a policy is admissible if all the Af belong to S .
When the policy has been chosen , the state of the product can be obtained from the state of the feed by repeated application of the transformation ( 1 ) ; ;
thus Af .
The objective function , which is to be maximized , is some function , usually piecewise continuous , of the product state .
Let this be denoted by Af .
An optimal policy is an admissible policy Af which maximizes the objective function P .
The policy may not be unique but the maximum value of P certainly is , and once the policy is specified this maximum can be calculated by ( 2 ) and ( 3 ) as a function of the feed state Af .
Let Af where the maximization is over all admissible policies Af .
When it is necessary to be specific we say that the optimal policy is an optimal R-stage policy with respect to the feed state Af .
For any choice of admissible policy Af in the first stage , the state of the stream leaving this stage is given by Af .
This is the feed state of the subsequent Af stages which , according to the principle of optimality , must use an optimal Af-stage policy with respect to this state .
This will result in a value Af of the objective function , and when Af is chosen correctly this will give Af , the maximum of the objective function .
Thus Af where the maximization is over all admissible policies Af , and Af is related to Af by ( 5 ) .
The sequence of equations ( 6 ) can be solved for Af when Af is known , and clearly Af , the maximization being over all admissible Af .
The set of equations ( 5 ) , ( 6 ) , and the starting equation ( 7 ) is of a recursive type well suited to programming on the digital computer .
In finding the optimal R-stage policy from that of Af stages , only the function Af is needed .
When Af has been found it may be transferred into the storage location of Af and the whole calculation repeated .
We also see how the results may be presented , although if n , the number of state variables , is large any tabulation will become cumbersome .
A table or set of tables may be set out as in Table 2.1 .
To extract the optimal R-stage policy with respect to the feed state Af , we enter section R of this table at the state Af and find immediately from the last column the maximum value of the objective function .
In the third column is given the optimal policy for stage R , and in the fourth , the resulting state of the stream when this policy is used .
Since by the principle of optimality the remaining stages use an optimal Af-stage policy with respect to Af , we may enter section Af of the table at this state Af and read off the optimal policy for stage Af and the resulting state Af .
Proceeding in this way up the table we extract the complete optimal policy and , if it is desired , we can check on Af by evaluating Af at the last stage .
It may be that the objective function depends not only on Af but also on Af , as when the cost of the operating policy is considered .
A moment's reflection shows that the above algorithm and presentation work equally well in this case .
A form of objective function that we shall often have occasion to consider is Af .
Here P represents the value of the stream in state P and Q the cost of operating the stage with conditions Q .
Hence P is the increase in value of the stream minus the cost of operation , that is , the net profit .
If Af denotes the net profit from stage R and Af , then the principle of optimality gives Af .
This sequence of equations may be started with the remark that with no process Af there is no profit , i.e. , Af .
2.3 the discrete stochastic process
The process in which the outcome of any one stage is known only statistically is also of interest , although for chemical reactor design it is not as important as the deterministic process .
In this case the stage R operating with conditions Af transforms the state of the stream from Af to Af , but only the probability distribution of Af is known .
This is specified by a distribution function Af such that the probability that Af lies in some region D of the stage space is Af .
We cannot now speak of maximizing the value of the objective function , since this function is now known only in a probabilistic sense .
We can , however , maximize its expected value .
For a single stage we may define Af where the maximization is by choice of Af .
We thus have an optimal policy which maximizes the expected value of the objective function for a given Af .
If we consider a process in which the outcome of one stage is known before passage to the next , then the principle of optimality shows that the policy in subsequent stages should be optimal with respect to the outcome of the first .
Then Af , the maximization being over all admissible Af and the integration over the whole of stage space .
The type of presentation of results used in the deterministic process may be used here , except that now the fourth column is redundant .
The third column gives the optimal policy , but we must wait to see the outcome of stage R and enter the preceding section of the table at this state .
The discussion of the optimal policy when the outcome of one stage is not known before passing to the next is a very much more difficult matter .
2.4 the continuous deterministic process
In many cases it is not possible to divide the process into a finite number of discrete stages , since the state of the stream is transformed in a continuous manner through the process .
We replace r , the number of the stage from the end of the process , by t , a continuous variable which measures the `` distance '' of the point considered from the end of the process .
The word distance is used here in a rather general sense ; ;
it may in fact be the time that will elapse before the end of the process .
If T is the total `` length '' of the process , its feed state may be denoted by a vector p(T) and the product state by p(Q) .
P denotes the state at any point T and Q the vector of operating variables there .