A factory makes three products called Spring, Autumn, and Winter, from three materials

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

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

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

each product is as follows:

a$)$ Formulate an LP model for the factory that maximises the profit, while satisfying the demand and the cotton and wool proportion constraints. There is no penalty for the shortage. $[20$ Marks$]$

b$)$ Solve the model using R$/$R Studio. Find the optimal profit and optimal values of the decision variables. $[20$ Marks$]$

Hints: You may refer to Week $5\mathrm{.}4$ Example $-$ Blending Crude Oils into Gasolines. For example, let xij $0$ be a decision variable that denotes the number of tons of products j $$ j in $\{1=$ Spring, $2=$ Autumn, $3=$ Winter$\}$ to be produced from Materials i in $\{$C$=$Cotton, W$=$Wool, S$=$Silk$\}.$