$4.$ A plant has been studying its maintenance load for the last three years. The load is found to be mainly of three types. Mechanical, electrical, and civil. The output of this study is to estimate the daily load. The load is divided in two priorities. Priority $1$ and priority $2.$ Priority $1$ requires highly skilled trade and priority $2$ can be executed by either highly skilled or regular trade. The cost of regular trader usually $70\%$ of the highly skilled trade. The hourly cost of highly skilled mechanical trade is $120$ Saudi Riyal $($SR$).$ The hourly cost of highly skilled electrical trade is $100$ SR and the cost of highly skilled civil trade is $80$ SR$.$ The daily demand for priority $1$ mechanical work is uniformly distributed with mean $50$ hours and lower and upper limits of $40$ and $60.$ The daily demand for priority $2$ mechanical work is distributed exponentially with a mean $100$ hours. The daily demand for priority $1$ electrical work is deterministic and equal $60$ and for priority $2$ is also deterministic and its value is $100$ hours. The daily demand for civil is all priority $2$ and its deterministic of value $40$ hours. The productivity of highly skilled trade for all specialties and type of work is $80\%$ and for regular trade is $70\%.$ The management wants priority $1$ to be completed with high reliability that exceeds $99\%$ and for priority $2$ the reliability should exceed only $80\%.$

$($i$)$ Formulate an optimization model to determine the optimal number of hired manhours to hire in each trade mechanical, electrical and civil taking care of the deterministic and random characteristics of the maintenance load. $(8)$

$($ii$)$ Transform the model you formulated in $($i$)$ in this question to obtain its certainty equivalent.$(4)$