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.

a) Formulate an LP model for the factory that maximises the prot, while satisfying the demand and the cotton and wool proportion constraints.

b) Solve the model using R/R Studio. Find the optimal prot and optimal values of the decision variables.

Hints: 1. Let xij 0 be a decision variable that denotes the number of tons of products j for j 2 f1 = Spring; 2 = Autumn; 3 = Winterg to be produced from Materials i 2 fC=Cotton, W=Wool, S=Silkg. 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 Spring is given by: xC1/ xC1 + xW1 + xS1 .

$5 Spring Autumn Winter Sales price Production cost $60 $55 $4 $60 $5 Cotton Wool Silk Purchase price $30 $45 $50 The maximal demand (in tons) for each product, the minimum cotton and wool propor- tion in each product is as follows: Spring Autumn Winter Demand min Cotton proportion min Wool proportion 4800 55% 30% 3000 45% 40% 3500 30% 50%