Three products are processed through three different operations. The following table gives the time required, profit per unit of each product and operation capacity:

$\backslash$table$[[$Operations$,$Time Per Unit $($Minutes $),\backslash$table$[[$Operation$],[$Capacity$],[$Minutes$/$Day$]]],[$Product I,Product II$,$Product III,$43],[1,3,4,3,46],[2,5,0,4,42],[3,3,6,2,],[$Profit$/$Unit $($$$),2,1,3,43]]$

The objective is to find product$-$mix which maximizes the profit.

$($a$)$ Formulate this as an L$.$P$.$ Problem.

$($b$)$ Convert the problem into dual form

$($c$)$ Use the simplex method to find optimal solution for the problem formulated in $($a$)$ above.

$6.$ A small city of $15,000$ people requires an average of $3$ litres of water daily. The city is supplied with water purified at a central water works, where water is purified by filtration, chlorination and addition of two chemicals softening chemical $x$ and health chemical Y$.$ Water works plans to purchase two popular brands of products, product A and product $B,$ which contain these two elements. One unit of product A gives $8Kg$ of $x$ and $3Kg$ of $Y.$ One unit of product $B$ gives $4Kg$ of $x$ and $9Kg$ of $Y.$ To maintain the water at a minimum level of softness and meet a minimum in health protection, it is decided that $150Kg$ and $100Kg$ of two chemicals that make up each product must be