$4.$ The Manager $($P&C$)$ was keen to avoid legal hassles in a specific contract and so he discussed at length with the contractor about the time required for various activities in the contract. The contract was of installing and commissioning a large, complex, special equipment in the manufacturing plant. The contractor was pushing his agenda forward and told the manager that he would do the best job in the world and so on and his best completion time for activities $1-2,2-4,2-3,3-5,4-5,3-6,4-6,$ and $5-6$ were $2,0\mathrm{.}5,1,1,6,1,3,4$ days respectively. The activities $2-3$ and $2-4$ could be completed only after the activity $1-2$ and the activity $3-5$ could be completed only after activities $1-2$ and $2-3.$ The activities $1-2,2-4$ and $2-3$ need to be completed before the activity $4-5.$ The activities $1-2$ and $2-3,$ and activities $1-2$ and $2-4$ need to be completed before the activities $3-6$ and $4-6$ respectively. Finally, activities $1-2,2-4,4-5,2-3$ and $3-5$ need to be completed before the activity $5-6.$ The Manager $($P&C$)$ had some data about this contractor and based on this data, the manager jotted down the time for various activities that he felt would be more realistic. The manager also added a column of worst$-$case scenario or maximum time required for completing each activity. Thus, the time required for activities $1-2,2-4,2-3,3-5,4-5,3-6,4-6,$ and $5-6$ in the realistic time column was $2,2\mathrm{.}5,1\mathrm{.}5,2\mathrm{.}5,7,2,4$ and $6$ and in the maximum time column was $8,7\mathrm{.}5,11,7,8,3,11$ and $8,$ respectively. The Manager $($P&C$)$ wanted to include a penalty if time for completion exceeded twenty days and at the same time the manager did not want to have legal hassles. Would the manager be right in suggesting a penalty beyond twenty days?

Kindly note the solution must be with the REGARDS TO PERT, so mathametical solving is required.

Thank you.