Power for Metropolis Hospital: Bidding for Generators

The Metropolis Health Center located in the city of Metropolis is a state$-$of$-$the$-$art hospital known for advanced equipment and intensive care units. The hospital aims to provide a reliable and high quality heal service. Among many major factors such as highly respected doctors, latest$-$technology equipments, well$-$train personnel etc., such a high quality target requires also minor factors to be well$-$planned and prepared. One these factors is$,$ unquestionably, an uninterrupted power source. Besides the energy requirement of the high$-$en equipments, in any hospital where continuous health care services are provided and various and lengthy surgic operations are performed, power outages pose a significant risk. Recent electricity blackouts in the city hav forced the hospital to upgrade its backup power system.

The hospital aims to establish a back$-$up system with three generators. When the mains power is cut of the primary generator comes into operation. If the primary generator fails, the secondary generator takes over and if that also fails, the tertiary generator comes into play. The failure of all three generators could lead tc a serious crisis, even resulting in loss of life for the hospital. Currently the hospital has only two generators, a two$-$years$-$old one and a six$-$years$-$old one.

The Metropolis Health Center was negotiating contract terms with a company $($will he referred as "the company"$)$ who approached the hospital to undertake the generator procurement and maintenance services for the next three years. The company must agree to the following conditions:

For the generators that are of the same age, periodic maintenance will be carried out simultaneously.

The supplier must guarantee that the probability of the hospital being out of electricity is less than $0\mathrm{.}0001.$

If all generators fail during a power outage, the supplier will be liable to pay a penalty of $$100$ million due to potential problems that might arise.

The company will be responsible for whether to renew the current machines, and will decide on the maintenance periods each year. $($Hint: There are three options, adding only a single new generator, replacing the six$-$years$-$old one and adding two generators, and replacing both current generators and adding three generators to the system.$)$

For the entire generator system to be replaced, there is a labor cost of $$20,000($independent of the number of generators replaced$).$ Each maintenance costs to the company $$50,000$ per generator $($Hint: You make, on average, $\frac{360}{t}$ maintenances per machine per year, $t$ being the time between maintenances.$).$ The procurement cost for each new generator is $$500,000.$ The company will be paid $$4,000,000$ per year for the next three years from this service.

The company is aware of the significance of the contract. Succeeding in this contract could assist the company in establishing connections with the government. Therefore, the company is willing to not make a profit from this project, may even embrace an expected loss of up to $$700,000.$ However, an amount larger than this would pose a significant risk to the company, and in such a case, the company would withdraw from contract negotiations.

To make an informed decision, the company has tasked its analysts with conducting a feasibility analysis on the matter. They are particularly interested in the probability of failure occurring $t$ days after a generator maintenance. The analysts have provided them with a report using historical data that shows the probability of failure during a power outage after $t^{th}$ day is $\frac{(k+1)(t^2+20t)}{50,000},k$ being the age of the machine $($the generator will be regarded as $0$ year old in the entire year it is bought, $1$ year old in the entire following year, $2$ years old the year after etc.. Similarly, current generators will be regarded as $2$ and $6$ years old respectively for the first year$).$

In the recent meeting, the hospital administration expressed frustration with the company spokesperson. Now, they want to make an auction and make you compete with another company. Even though the company is willing to accept any condition demanded by the government, the probability of winning the contract is $50\%.$ To enter the auction, the company needs to pay $$500,000$ fee in advance. Is bidding profitable? What is the expected loss due to unsettling the administration?