A factory makes three products called Bloom, Amber, and Leaf, from three materials

containing Cotton, Wool and Nylon. The following table provides details on the sales

price, production cost and purchase cost per ton of products and materials respectively.

Sales price Production cost Purchase price

Bloom $$60$ $$5$ Cotton $$40$

Amber $$55$ $$4$ Wool $$45$

Leaf $$60$ $$5$ Nylon $$30$

The maximal demand $($in tons$)$ for each product, the minimum cotton and wool proportion in each product are as follows:

Demand min Cotton proportion min Wool proportion

Bloom $420050\%40\%$

Amber $320060\%40\%$

Leaf $350050\%30\%$

a$)$ Formulate an LP model for the factory that maximises the profit, while satisfying the

demand and the cotton and wool proportion constraints.

$[20$ Marks$]$

b$)$ Solve the model using R$/$R Studio. Find the optimal profit and optimal values of the

decision variables.

$[20$ Marks$]$

Hints:

$1.$ Let xij $>=0$ be a decision variable that denotes the number of tons of products j for

j in $\{1=$ Bloom, $2=$ Amber, $3=$ Leaf$\}$ to be produced from Materials i in $\{$C$=$Cotton,

W$=$Wool, N$=$Nylon$\}.$

$2.$ The proportion of a particular type of Material in a particular type of Product can be

calculated as:

e$.$g$.,$ the proportion of Cotton in product Bloom is given by: xC$1$

xC$1+$ xW$1+$ xN$1$