Please solve below questions and give correct option. Solve the numericals in details

$1)$ Consider a dataset with the following data points: $(2,3),(4,4),$ and $(6,7).$ Perform k$-$means clustering with k$=2,$ starting with initial centroids at $(2,3)$ in cluster $1$ and $(6,7)$ in cluster $2.$ Calculate the final centroids after convergence.

$(2,3)$ and $(6,7)$

$(2,3)$ and $(4,4)$

$(4,4)$ and $(6,7)$

$(3,3\mathrm{.}5)$ and $(6,7)$

$2)$

Now consider the point $(5,6),$ which cluster will it belong with respect to the Q$4?$

Cluster $1$

Cluster $2$

Both the clusters

None of them

$3)$What is the typical approach for determining the optimal value of $$k$$ in k$-$means clustering?

Trial and Error

Elbow method

Eucledian method

Randomly

$4)$

If we have multiple inputs and outputs, the efficiency score is calculated as the ratio of:

Outputs to inputs

Inputs to outputs

Weighted sum of outputs to inputs

None

$5)$

A manufacturing company produces two products, A and B$,$ using two inputs, labor and raw materials. The efficiency frontier is constructed based on the production of these two products. If the company produces $500$ units of product A and $300$ units of product B using $100$ units of labor and $200$ units of raw materials, and it is operating on the efficiency frontier, what is its efficiency score?

$6)$

Given three facilities A$,$ B$,$ and C with input values as $3,5,$ and $4$ and output values as $4,5,$ and $3$ respectively. Calculate the efficiency score of facility B$.$

$7)$

Which facilities lie on efficiency frontier with respect to Q$9?$

A and B

B and C

Only A

Only B

$8)$

As facility $8$ is the least efficient among all, what is the percentage increase in efficiency if we increase OEE by $5$ units? $($Hint: You may need to use excel solver$)$

Considering the same output table, which parameter should we focus on to increase the efficiency of facility $2?$

Production Yield

OEE

Cycle time

Resource Utilization

Consider the Constraint $0,$ if the right$-$hand side of the constraint is changed to a constant $$k$($instead of $1),$ then what is the maximum possible value of the objective function $($theoretically$)?$

k

k$+1$

$1/$k

Remains same