A company manufactures three models of cars. There is a backlog of orders with the company. Model A requires $167,167,$ and $178$ worker$-$days in three production processes I, II$,$ and III, respectively. Model B requires $235,398,$ and $198$ worker$-$days, while Model C requires $278,341,$ and $253$ worker$-$days respectively, in the three production processes. The number of workers employed in the three production processes are $20,34,$ and $32,$ respectively, and an average worker is on the job for $200$ working days in a year. The expected profit per car of the three models is Rs$.7928,$ Rs$.17098,$ and Rs$.32813$ respectively. $($i$)$ Determine the optimal product$-$mix and the total maximum profit obtainable. $($ii$)$ What are the shadow prices of the resources? $($iii$)$ Write the dual to the above LPP