Assignment $(20\backslash \%)$You are in charge of a refurbishment work project of a faculty building to create upgraded spaces and environment to accommodate the upcoming intake of students. A delay in the project will cause a number of losses including cost overruns and negative impact on the reputation. The time and cost information for the project are listed in Table $1.$ A variable overhead cost of $\backslash (\backslash$$ $2500\backslash )$ per day is applied for the project. A penalty cost of $\backslash (\backslash$$ $15000\backslash )$ per day will be imposed on day $19$ and beyond. Assume the project is started on $7$ January $2025,$ what would be your target completion date and the total project costs saved if your interest is to minimize the cost of the project?Table $1\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|$c$|\}\backslash$hline Activity & $\backslash$begin$\{$tabular$\}\{$c$\}$ Immediate $\backslash \backslash$Predecessors$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$ Normal $\backslash \backslash$Time $($days$)\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$ Normal $\backslash \backslash$Cost $(\backslash$$$)\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$ Crash Time $\backslash \backslash ($days$)\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$ Crash Cost $\backslash \backslash (\backslash$$$)\backslash$end$\{$tabular$\}\backslash \backslash \backslash$hline A & $-$ & $3$ & $10000$ & $2$ & $15000\backslash \backslash \backslash$hline B & $-$ & $4$ & $15000$ & $3$ & $18000\backslash \backslash \backslash$hline C & A & $2$ & $12500$ & $1$ & $16000\backslash \backslash \backslash$hline $\backslash (\backslash$mathbf$\{$D$\}\backslash )$ & B & $5$ & $17500$ & $3$ & $21500\backslash \backslash \backslash$hline $\backslash (\backslash$mathbf$\{$E$\}\backslash )$ & B & $3$ & $15000$ & $3$ & $16000\backslash \backslash \backslash$hline F & C$,$ D & $4$ & $20000$ & $2$ & $25000\backslash \backslash \backslash$hline $\backslash (\backslash$mathbf$\{$G$\}\backslash )$ & C & $6$ & $7500$ & $3$ & $15000\backslash \backslash \backslash$hline $\backslash (\backslash$mathbf$\{$H$\}\backslash )$ & C$,$ D$,$ E & $2$ & $5000$ & $1$ & $6000\backslash \backslash \backslash$hline $\backslash (\backslash$mathbf$\{$I$\}\backslash )$ & E & $3$ & $10000$ & $1$ & $16000\backslash \backslash \backslash$hline J & D$,$ E & $4$ & $7500$ & $3$ & $10000\backslash \backslash \backslash$hline K & F$,$ G & $3$ & $15000$ & $1$ & $20000\backslash \backslash \backslash$hline $\backslash (\backslash$mathbf$\{$L$\}\backslash )$ & G$,$ H$,$ I & $3$ & $15000$ & $2$ & $19000\backslash \backslash \backslash$hline $\backslash (\backslash$mathbf$\{$M$\}\backslash )$ & I, J & $2$ & $10000$ & $1$ & $12000\backslash \backslash \backslash$hline $\backslash (\backslash$mathbf$\{$N$\}\backslash )$ & K$,$ M & $3$ & $17500$ & $2$ & $18500\backslash \backslash \backslash$hline $\backslash (\backslash$mathbf$\{$O$\}\backslash )$ & H$,$ M & $1$ & $20000$ & $1$ & $22000\backslash \backslash \backslash$hline $\backslash (\backslash$mathbf$\{$P$\}\backslash )$ & N$,$ L$,$ O & $5$ & $15000$ & $4$ & $20000\backslash \backslash \backslash$hline$\backslash$end$\{$tabular$\}$The new critical path$($s$)$ has been identified after performing activities crashing. What is the probability that these path$($s$)$ will only be completed after the Chinese New Year $2025$ if without implementing the crash cost? Find the date by which $99\mathrm{.}9$ percent of the project would be completed by then. Table $2$ shows the time estimates for each activity of the project.Table $2\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|\}\backslash$hline $\backslash$multirow$\{2\}\{*\}\{$Activity$\}$ & $\backslash$multicolumn$\{3\}\{|$c$|\}\{$Time Estimates $($days$)\}\backslash \backslash \backslash$hline & Optimistic & Most Likely & Pessimistic $\backslash \backslash \backslash$hline A & $2$ & $3$ & $5\backslash \backslash \backslash$hline B & $3$ & $4$ & $5\backslash \backslash \backslash$hline C & $2$ & $2$ & $2\backslash \backslash \backslash$hline D & $3$ & $5$ & $6\backslash \backslash \backslash$hline E & $2$ & $3$ & $4\backslash \backslash \backslash$hline F & $3$ & $4$ & $6\backslash \backslash \backslash$hline G & $3$ & $6$ & $10\backslash \backslash \backslash$hline H & $1$ & $2$ & $3\backslash \backslash \backslash$hline I & $1$ & $3$ & $5\backslash \backslash \backslash$hline J & $2$ & $4$ & $6\backslash \backslash \backslash$hline K & $2$ & $3$ & $5\backslash \backslash \backslash$hline L & $1$ & $3$ & $4\backslash \backslash \backslash$hline M & $0$ & $2$ & $4\backslash \backslash \backslash$hline N & $1$ & $3$ & $6\backslash \backslash \backslash$hline O & $0$ & $1$ & $2\backslash \backslash \backslash$hline P & $2$ & $5$ & $8\backslash \backslash \backslash$hline$\backslash$end$\{$tabular$\}-$ QUESTION END $-$