Consider a cutting$-$stock problem where an unlimited supply of rolls of length $27$ is available. Three smaller

types of rolls of sizes $7,9$ and $12$ are required to be produced to satisfy a demand of $30,25$ and $15$ rolls,

respectively for each kind. Let us denote $($a$,$ b$,$ c$)$ a cutting pattern where a rolls of size $7,$ b rolls of size $9$ and c

rolls of size $12$ are produced from a roll of length $27,$ such that $7$a $+9$b $+12$c $<=27.$

$1.$ Write the associated master problem $($MP$)$ initialized with the cutting patterns $(3,0,0),(0,3,0),(0,0,2)$

$2.$ Write the associated subproblem $($SP$),$ assuming that we denote $\backslash$alpha $7,\backslash$alpha $9,\backslash$alpha $12$ the dual variable associated

with the demand constraint associated with every small roll

$3.$ Find the optimal solution to the linear relaxation of the MP by column generation. Solve the associated

linear or integer problems