Project$/$Event Management Case $($Show all your works: including all computational procedures and details.$)$

The Rare Natural Resources Development $($RNRD$)$ Corporation is undertaking a Deep$-$Sea Oil Field Exploration $($DOED$)$ project. RNRD undertakes $360$ DOED projects annually. The outcome of the DOED project will either result in a successful oil field development or in failure. The DOED project typically explores two types of oil fields: one is the exploration of small, nearshore oil fields, and the other is the exploration of large, deep$-$sea oil fields. Approximately $1/3$ of RNRD$\text{'}$s projects consist of large oil fields, while the remaining $2/3$ are focused on smaller$-$scale developments. The exploration process begins with a preliminary inspection of the oil field. This initial inspection assesses the likelihood of successful development. Oil fields deemed likely to succeed proceed to full$-$scale exploration, while those assessed as likely to fail are abandoned. This preliminary inspection process, however, is not highly accurate for either large or small oil fields and carries the risk of misjudgment. Regardless of the size of the oil field project, approximately $30\%$ of oil fields are deemed likely to fail, and in these cases, the failure rate is inevitably $100\%.$ Of the oil fields deemed likely to succeed $(70\%),2/7$ are later found to be failures during full$-$scale exploration process, while $5/7$ ultimately achieve successful development. The preliminary inspection of an oil field, regardless of its size, typically takes $1$ month to complete. Full$-$scale exploration spans $5$ months for small oil fields and $8$ months for large ones. A successful development yields $$8$ million in revenue for a small oil field and $$10$ million for a large one. Conversely, a failed project generates no revenue, regardless of the field's size. Preliminary inspections require $2$ engineers, regardless of the oil field size, and incur $$200,000$ per month in utility and resource costs. Full$-$scale exploring a small oil field requires $5$ engineers, while a large field requires $12.$ Each engineer earns a monthly wage of $$12,000.$ Additionally, small oil field exploration incurs $$600,000$ per month in utility and resource costs, while large field exploration costs $$1,200,000$ per month.

$($a$)$ Evaluate the$$

Monthly Profit$=$Total Revenue$$ Monthly Wage$$ Monthly Utility and Resource

$($b$)$ Evaluate the following two options and choose the optimal option $($assuming their investment costs are identical$)$

OPTION $1.$ By purchasing new equipment, the exploration period for a large field is reduced by $0\mathrm{.}1$ months.

OPTION $2.$ By increasing research efforts, the annual project volume will rise from the current $360$ projects to $500.$