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Maternity Ward Capacity and Oil Exploration: Case Studies
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## Maternity Ward Capacity Management

The maternity ward of a large London-based hospital trust has signed a contract with the National Health Service to deliver 14000 babies in 2021. To achieve this, it needs to have approximately 270 deliveries per week (=14000 deliveries/52 weeks). Deliveries are booked when the expecting mother is towards the end of the first trimester of gestation (i.e., about 6 months before they actually take place). However, one concern is the phenomenon of no-shows  women who book but never show up to deliver. This happens for a number of reasons, e.g., the mother moves out of London, experiences a miscarriage, etc. In the past, on weeks where 270 deliveries were scheduled, approximately 12% of the booked deliveries did not show up, but this number varies from week to week it could be as low as 6% or as high as 18%.

a. (5 points) What would be a good distribution to choose to model the number of mothers booked per week who do not show up to deliver? Justify your choice.

Given the 12% no-show rate, it has been argued that if the hospital wants to meet the target of 14000 deliveries per year, it will need to book an additional 12% deliveries, i.e, instead of booking 270 deliveries per week it will need to book 303 (an extra 33 =12%*270 bookings per week). This will ensure that it will not miss the 14000 deliveries target.

b. (5 points) What do you think of this reasoning?

c. (5 points) How would you build a simulation model to estimate the probability of achieving the 14000 births target if the number of deliveries booked per week is 307?

Figure 1 below shows the histogram, the cumulative distribution function and summary statistics of a simulation of the number of deliveries per year when the number of weekly bookings is 307. This is based on a simulation model and 10,000 simulation runs.

 Min 1st Quartile Median Mean 3rd Quartile Max SD 13792 13997 14044 14043 14089 14317 68.42
 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 13929 13954 13971 13985 13997 14007 14017 14026 14035 14043 55% 60% 65% 70% 75% 80% 85% 90% 95% 14051 14060 14069 14079 14088 14100 14113 14130 14155

Figure 1: Histogram, cumulative distribution function and summary statistics of the number of deliveries per year when the number of weekly bookings is 307.

d. (5 points) Use these figures to interpret the results of the simulation. Can you estimate the probability that the hospital trust meets the 14000 annual target?

RiskyOil Inc. is considering the undertaking of a new venture. The government of the Mediterranean island of Mepos has decided to put the highly risky Poseidon plot up for leasing for oil exploration and production. RiskyOil is the only company considering bidding for the lease. To even consider the bid, the Mepos government requires an upfront fee of \$6M. If the bid is successful, which will be announced shortly after the payment of the upfront fee, the government will grant permission to RiskyOil to begin exploration and extraction immediately; if it is not successful, the \$6M upfront fee will be lost.

## Question 1: No-Show Phenomenon and Simulation Model

Olive, the exploration and production director of RiskyOil, believes that, if RiskyOil decides to bid, the probability that the bid will be successful is 50%. The oil reserves have been mapped and are known to be worth \$120M (after RiskyOil pays the Mepos government royalties and extraction fees). The precise costs of extracting the oil are not known at the time of the bidding decision; they will become known after the bid is successful and before extraction begins. From past experience with such projects, Olive believes costs will either be \$144M with probability 70%, or \$72M with probability 30%. In either case, RiskyOil does not have to extract the oil and can always return the plot to the government for no extra cost.

Olive is considering whether this is a project worth bidding for.

a. Frame the problem. What are Olive s objectives? Lay down the decisions she has to take and the uncertainties she faces in chronological order. (5 points)

b. Draw a decision tree to determine the payoffs and risks associated with each decision. Use the tree to help Olive make her decisions. You can do this by hand or using Precision Tree.(10 points)

Below you can see the results of two sensitivity analyses. In the first, you can see the value of the decision to bid for the lease as the probability that the lease is granted changes from 0 to 1 (currently this probability is estimated to be 50%). In the second you can see the value of the lease as the probability that the costs of extraction are high varies from 0 to 1 (currently this probability is estimated to be 70%). In both sensitivity analyses the value of not biding for the lease is shown in red.

c. Interpret the sensitivity analyses. Based on these, what recommendations would you give to Olive? (5 points)

In a conversation with Yiangos, the head of submarine exploration, Olive was informed that RiskyOil might be able to deploy a newly developed space-based exploration test in order to assess the extraction costs before deciding whether to bid or not for the lease. The test could be deployed immediately (ie., before RiskyOil decides if it wants to bid for the lease) at a cost of \$1.4M. The test is believed to be 100% accurate.

d. What is the value of the test? Draw a new decision tree to help Olive decide if she should follow this advice. (5 points)

On further scrutiny, Olive discovered that unfortunately the space test is not 100% accurate. In the past, when the costs where high, it had correctly predicted high costs 80% of the time and incorrectly predicted low costs 20% of the time. In contrast, when the costs where low, it had correctly predicted the costs would be low 90% of the time and 10% of the time it had incorrectly predicted costs as high.

e. Does this change the value of the test? Is it still worth pursuing? (5 points)