a$)$ Explain the steps followed in formulating a standard linear program problem $(6$marks$)$

b$)$ A firm produces three products $\backslash ($ X$,$ Y$,$ Z $\backslash )$ with a contribution of ksh$\mathrm{.}20,18$ and $16$ respectively. Production data is as follows:

$\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|$l$|\}$

$\backslash$hline & Machine Hours & Labour Hours & Materials $\backslash \backslash$

$\backslash$hline$\backslash ($ X $\backslash )$ & $5$ & $2$ & $8\backslash \backslash$

$\backslash$hline$\backslash ($ Y $\backslash )$ & $3$ & $5$ & $10\backslash \backslash$

$\backslash$hline$\backslash ($ Z $\backslash )$ & $6$ & $3$ & $3\backslash \backslash$

$\backslash$hline Availability & $3000$ & $2,500$ & $10,000\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

The manufacturer wishes to establish the weekly production plan which maximizes contributuion using the simplex method $(10$ marks$)$

c$)$ Explain and illustreate the concept of Shadow prices $(4$ marks$)$

d$)$ A Manufucturer is to market a new fertilizer which is to be a mixture of two ingredients A and B$.$ The properties of the two Ingredients are: