A discrete return function is a vector values which represent the return for the level of investment, effort, etc a measured by the index of positions.
The Index value is chosen at the appropiate scale, i.e say increments of $10,000 for a series of maintenance, repair, replacement and capital expansions.
For particular projects the return function eventually does not rise for increased levels of effort and when this happens, additional effort should cease.
In the case of investment, the rule usually is to invest only until the yield on the next increment is less than the minimum acceptable rate of return.
There are no reasons why the low end of investment should yield a positive retun, but eventually a level should be reached where the return is positive and investment can be continued until one of the above rules is violated.
Dynamic programming states: " An optimal policy has the the property that, whatever the inital state, and inital decisions are, the remaining decisions must constitute an optimal policy with reguard to the state resulting from the first decision.
The primary argument in favor of dynamic programming is avoiding the difficulties associated with the classical variational approach of the calculus.
Dynamic programming provides a search process which is more efficient than enumeration.
Wagner states: "Because no satisfactory canonical form exists for all types (not even for finite, discrete, deterministic models) there is no single efficient computer program to handle all dynamic programming problems."
The form of the DPR is usually simple enough so that it is usuall not much trouble to write a program for a particular problem.
The dynamic programming approach used in the ws OPT is to combine pairs of return functions to produce a third or composite which is as good or better than best of the two.
The composite return function is an orderd vector which gives the best return for all possible combinations of the parts of the two original vectors. It can be truncated at a predetermined budget level by selecting the number of levels equal to the budget.
The result of function A DP1 B is a 3 x n array, row 1 gives the composite value, row 2 the index A, and row 3 the index of B combined to produce the composite.
To combine a number of projects each with a multilevel return function the proceedure is pair the first with the second to produce a composite which is paired with the third to produce another composite, and so on until all the possible levels of all the alternative have been considered.
The budget limit can be imposed at any stage or left until the grand composite is determined. To determine which level of which alternative should be chosen it is necessary to work back from the final composite to the first noting which combinations have been used to determine each level.
Example return functions from: Fabrycy, p426 v15 15 40 80 90 95 98 100 v25 15 40 60 70 73 74 75 v34 26 40 45 50 51 52 53 ‘vs v1, [1] v2, [.1] v3
End to date, ams 990712