The following problem is adapted from Sathaye and Hall (1976). A long aqueduct is to be si...

The following problem is adapted from Sathaye and Hall (1976). A long aqueduct is to be sized by sections to deliver irrigation water to n irrigation districts, as shown in the sketch below. The amount of water to be used by each district is to be determined

Define:

xk = amount of water delivered annually to irrigation district k;

Q = total amount of water available;

qk = total amount of water allocated to the first k districts;

vk(xk)= value or net benefit (present worth) from allocating an amount of water Xk to irrigation district k; and

ck(qk)= cost (present worth) of aqueduct section k, based on an annual total flow of qk.

It is desired to maximize overall net benefits of the project.

(a) Write the mathematical programming model for the problem.

(b) Write the dynamic programming recursive equation that will solve the problem.

(c) If 1,00,000 acre-feet of water were available (equal to Q) xk and qk were broken into 100,00 AF increments, and n =5 districts:

(1) How many possible solutions are there (by enumeration)?

(2) How many allocations will have to be compared using DP?