Homework Problem: The water company wants to know how much water it will need to

Question

Homework Problem: The water company wants to know how much water it will need to supply. On average, the lawns in this city need 1 million gallons per day. But, if it rains, the rain might supply 0-2 million gallons per day. On average, rain in the drought supplies 5 million gallons per month. Rain at a rate of more than what the lawns need runs off into the sewer and cannot be reclaimed or used.

Part 1A: Use this information to produce a simulation to determine how much the water utility must purchase for the next 3 months. The simulation must be user-friendly.

Part 1B: Run your simulation a number of times and interpret the statistics. How much should the utility purchase each month to guarantee a service level of 90%?

Part 2A: The town is stepping up enforcement of watering violations. 90% of the people each month who violate watering restrictions get away with it (Freebirds). Of those who get caught (Jailbirds), 80% won't do it again (and become Angels). Angels are rarely tempted to violate the restrictions next month (10%). Build a Markov Process model using these three states and given 50% of the town is estimated to initially be Angels, is the policing effective? What percentage of the population violates restrictions after 10 months.

Part 2B: Run your Markov model to determine which is more effective: Policing to catch 20% of the violators each month or public service announcements to reduce the Angel temptation rate to 3%.

Explanation / Answer

Let us calculate the water needed for three months.
Number of days in three months = 91 days
Thus, the total requirement of water for the three month period= 91 days * 1million gallon per day = 91 million

Now, the water supplied by rain in the three months
3 months * 5million gallons per month = 15 million gallons

Now, this rain amount per day can be 0,1 or 2 million gallons per day.
Now, on those days on which there is a rainfall of 2 million gallons, only 1 million is used and rest is wasted away and only 1 million gallon is used.

The various possibilities with rain amount are as follows

Case 1
15 days of 1 million gallons/day and 0 day of 2 million gallons/day

Water Requirement = 91 million - 15days*1million/day = 76 million

Case 2
13 days of 1 million gallons/day and 1 day of 2 million gallons/day

Water Requirement = 91 million - (13+1)days*1million/day = 77million gallons

Case 3
11 days of 1 million gallons/day and 2 day of 2 million gallons/day

Water Requirement = 91 million - (11+2)days*1million/day = 78million gallons

Case 4
9 days of 1 million gallons/day and 3 day of 2 million gallons/day

Water Requirement = 91 million - (9+3)days*1million/day = 79million gallons

Case 5
7 days of 1 million gallons/day and 4 day of 2 million gallons/day

Water Requirement = 91 million - (7+4)days*1million/day = 80million gallons

Case 6
5 days of 1 million gallons/day and 5 day of 2 million gallons/day

Water Requirement = 91 million - (5+5)days*1million/day = 81million gallons

Case 7
3 days of 1 million gallons/day and 6 day of 2 million gallons/day

Water Requirement = 91 million - (3+6)days*1million/day = 82million gallons

Case 8
1 days of 1 million gallons/day and 7 day of 2 million gallons/day

Water Requirement = 91 million - (1+7)days*1million/day = 83million gallons

1B)
Now, assuming each of the cases to be equally probable, we have eight equally probable cases.
Thus, even if we exclude one case in our consideration, we will have a service level
=(8-1)/8 *100% = 87.5%
So, we have to plan our purchase according to the Case with highest water requirement.

Thus, Water requirement must be per month that along the three months period, we get 83 million gallon of water.
This gives us the per month value
= 83million gallons / 3 months = 27.67 million gallons/ month

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