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.

Sales price Production cost Purchase price

Spring $$60$ $$5$ Cotton $$30$

Autumn $$55$ $$3$ Wool $$45$

Winter $$60$ $$5$ Silk $$50$

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

tion in each product is as follows:

Demand min Cotton proportion min Wool proportion

Spring $360055\%30\%$

Autumn $330045\%40\%$

Winter $400030\%50\%$

a$)$ Formulate an LP model for the factory that maximises the prot, 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 prot and optimal values of the

decision variables.

$[20$ Marks$]$

Hints:

You may refer to Week $8\mathrm{.}7$ Example $-$ Blending Crude Oils into Gasolines. For ex$-$

ample, let xij $0$ be a decision variable that denotes the number of tons of products

j for j $2$ f$1=$ Spring; $2=$ Autumn; $3=$ Winterg to be produced from Materials

i $2$ fC$=$Cotton, W$=$Wool, S$=$Silkg.