Consider the case of parallel streams to be used for water supply for a city. Assume that the best site for a reservoir has been found on each stream and that the cost has been evaluated in terms of reservoir capacity at each site. Further, for each site the safe yield (based on a once-in-50-year recurrence interval shortage) has been found as a function of reservoir capacity. Cost call then be expressed in terms of safe yield for each potential reservoir site as shown in the accompanying table. The city wants to meet its water supply need at least cost. Further development of this problem can be found in Walhne et al. (1975).
Define:
xk = safe yield draft from eservoir k (on stream k);
ck(xk) = cost as function of safe yield draft from reservoir k;
D = total water supply draft reqired by the city; and
n = number of parallel streams.
(a) Write the mathematical programming model that describes this problem.
(b) Write the dynamic programming recursive equation for the problem.
(c) Construct tables and calculate the optimal cost and associated optimal draft from each stream to achieve a total draft of from 30 to 100 million gallons per day (MGD) in increments of 10MGD.
(d) Plot optimal cost versus total safe yield
Stream | Safe yield Draft (MGD) | Reservoir Total Cost ($ million) |
1 | 10 | 0 |
| 20 | 22 |
| 30 | 44 |
2 | 20 | 0 |
| 30 | 6 |
| 40 | 18 |
| 50 | 50 |
3 | 0 | 0 |
| 10 | 10 |
| 20 | 14 |
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