2.A manufacturing company produces r units of product 1 and z2 units of product2 per day.Product 1 makes a profit of￡l0 per unit,and product2 makes a loss of￡I per unit.The total combined demand for the two products is 500 units per day,and the totaldemand for product 2 specifically is 100 units per day;the company must produceenough to meet this demand.One unit of product 1 uses 1 kg of resource A and 4 g of resource B;one unit ofproduct 2 uses 3 kg of resource A and 1 g of resource B.There are a total of 2100kg of resource A and 2200 g of resource B available each day.This leads to the company attempting to solve the linear programming problem:Maximise z=10z1-2subject to 1+2 2500units;total demand22100units;specific demand for product 2×1+3×2≤2100kg:resource A4r1+x2≤2200g;resource BT1,x220.(a)Solve the linear programming problem using the graphical solution method.and report the optimal number of units of products 1 and 2 to produce eachday,along with the profit.(5 marks(b)A global shortage occurs in resource B,which means that only(22O0-△)gare available each day(for some△2o).At the same time,this shortageincreases demand,because your competitors are unable to produce enough oftheir own products.This leads to the total demand for the products increasingto50+2△.(The specific demand for product2 remains unchanged,asdothe profits per unit product.)Using your graphical solution,determine how your profit changes per unit△(for△comparatively small),and for what range of△this is valid.[5 marks](c)Solve the linear programming problem again,this time using the two-phasesimplex method.(Hint:at some point,you may have a choice between entering r or enteringr4.It will be faster to enter r1.)(5 marks)(d)By considering carefully how our method for calculating shadow prices extendsto this situation,answer (b)using your simplex tableaux,without any referenceto the graphical solution.You must explain your method.[5 marks]
