DecorIt manufactures wallpaper rolls for house decoration. The length of each roll must be customized

according to the needs of the customers $($retail stores all over the country$).$

In general, the length of the ordered rolls can be either $8$ meters, $15$ meters, $25$ meters or $30$ meters.

DecorIt purchases $30$ meters rolls from an outside supplier and then cuts the rolls as needed for their products.

Each $30$ meters roll can be used to make a number of smaller rolls. For example, a $30$ meters roll could be

used to make a $30$ meters roll $($with no waste$)$ or a $15$ meter roll and an $8$ meter roll $($with $7$ meters waste$)$ or

three $8$ meter rolls $($with $6$ meters waste$),$ etc. Appendix A shows all the possibilities for cutting a $30$ meters

roll into $8,15,25$ and $30$ meter rolls.

Every week, DecorIt needs $30$ of the $8$ meters rolls, $40$ of the $15$ meters rolls, $30$ of the $25$ meters rolls and $50$

of the $30$ meters rolls.

$1.$ What is the fewest number of $30$ meters rolls that can be purchased to meet their production needs?

$($a$)$ Formulate and explain an integer programming model in algebraic form for this cutting stock problem.

$(4$ marks$)$

$($b$)$ Formulate and solve the model in a spreadsheet for this problem. $(5$ marks$)$

$($c$)$ Discuss the optimal solution obtained and evaluate the amount of material waste for this solution.

$(2$ marks$)$

$2.$ The current price of a $30$ meters roll is $100$$$.$ The supplier of rolls proposed to provide $50$ meters rolls at

the price of $190$$ per roll. Should DecorIt accept this proposal or not? $(4$ marks$)$

$($a$)$ Identify all the different ways of cutting up a $50$ meters roll into $8,15,25$ and $30$ meter rolls.

$($b$)$ Formulate and solve the model in a spreadsheet for this new problem and based on the change in total

optimal cost, make a recommendation to accept the supplier proposal or not.

Appendix A

$\backslash$table$[[,$Pattern$],[$Length of Cut,$1,2,3,4,5],[8,1,3,,,],[15,1,,2,,],[25,,,,1,],[30,,,,,1]]$