A company is weighing the decision between acquiring a new or used production machine, with plans to sell the machine in two years upon completion of the current production contract. The new machine comes with a $$95,000$ price tag and incurs $$450$ monthly for maintenance and repairs. On the other hand, the used machine, priced at $$55,000,$ carries uncertain maintenance costs $$ a $25\%$ chance of $$1,400$ per month and a $75\%$ chance of $$1,100$ per month. Anticipating the machine's resale value in two years, the company expects the new machine to be valued at $$45,000,$ while the used machine is estimated to be worth $$25,000.$

As an alternative, the company can opt not to purchase a machine and instead outsource their production to A Industries, equipped with the necessary machinery. The upfront sub$-$contract fee stands at $$85,000.$ However, there is a potential risk associated with A Industries' historical performance $$ a $10\%$ probability of non$-$delivery. In the event of non$-$delivery, the company would incur a substantial penalty of half a million dollars for failing to fulfill the contract. Conversely, if A Industries delivers on time, the arrangement proceeds without issues.

$1.$ Represent the decision$-$making scenario for the CEO using an influence diagram. Include a table alongside a decision node to outline the alternatives and associated values.

$2.$ Develop a decision tree for the CEO's decision, ensuring that the appropriate numerical values are placed at relevant points in the tree. Clearly outline all the strategies in the decision tree.

$3.$ Generate risk profiles and cumulative risk profiles for each possible strategy in the figure. Determine if one strategy stochastically dominates the others and explain your conclusion with a simple "Yes" or $"$No$."$

$4.$ Assume uncertainty regarding the salvage value of the used machine after two years. Conduct a one$-$way analysis on the salvage value for the used machine and compare it with other strategies. Clearly present both graphs and numeric details for the analysis, and specify the assumed low and high ranges for the salvage value.

$5.$ Create a tornado diagram featuring the six parameters: purchase prices for new and used machines, monthly maintenance fees for new and used machines, and salvage values for new and used machines.

$6.$ Perform a sensitivity analysis on the probability that A Industries may fail to deliver at the end of the sub$-$contract. Compare all strategies based on expected monetary value $($EMV$)$ and identify the preferred strategy within specific probability ranges.