Jordao OY refines ferrochrome and nickel. It is currently using its supplies of these metals to...

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Jordao OY refines ferrochrome and nickel. It is currently using its supplies of these metals to produce standard alloy mixes for specialist steel makers. It is making two kinds of mix: General Purpose Stainless (GPS) and High Moly (HM). The two mixes contain four different metals: ferrochrome and nickel (produced in its own refinery) and molybdenum and cobalt (which it buys in the market). The costs of each of the four metals and the exact amounts needed for one tonne of each kind of mix are as follows: GPS Tonnes/ HM Tonnes/ Cost /tonne tonne of GPS tonne of HM Ferrochrome 0.50 0.21 2,170 Nickel 0.18 0.24 10,080 Molybdenum 0.27 0.51 11,070 Cobalt 0.05 0.04 27,630 Total 1.00 1.00 The total amount of ferrochrome that is available this month to make GPS and HM is 1,050 tonnes. The amount of nickel that is available is 810 tonnes. There is a constraint on production capacity which means that the total combined tonnage of GPS and HM that can be produced this month is not more than 3,600 tonnes. This month Jordao can sell the GPS mix for 7,460/tonne and the HM mix for 9,830/tonne. (i) Formulate a linear program to determine how much of each of the two mixes to make this month so as to maximise profit (define any variables you use, and show what any functions or inequalities represent in your formulation). [5 marks] (ii) Solve the problem in (i) graphically. Support any numerical results with calculations. [8 marks] (iii) (a) Adjust your result in (ii) on the assumption that capacity increases from 3,600 tonnes of the mixes to 3,601 tonnes this month. [4 marks] (b) Hence, find the shadow price of additional production capacity. [2 marks] (c) Calculate the maximum amount to which production capacity can be increased with the same shadow price. [3 marks] (d) - Question continues - Page 3 of 7 Show that the new profit for the capacity calculated in (c) corresponds to the shadow price calculated in (b). [3 marks] Jordao OY refines ferrochrome and nickel. It is currently using its supplies of these metals to produce standard alloy mixes for specialist steel makers. It is making two kinds of mix: General Purpose Stainless (GPS) and High Moly (HM). The two mixes contain four different metals: ferrochrome and nickel (produced in its own refinery) and molybdenum and cobalt (which it buys in the market). The costs of each of the four metals and the exact amounts needed for one tonne of each kind of mix are as follows: GPS Tonnes/ HM Tonnes/ Cost /tonne tonne of GPS tonne of HM Ferrochrome 0.50 0.21 2,170 Nickel 0.18 0.24 10,080 Molybdenum 0.27 0.51 11,070 Cobalt 0.05 0.04 27,630 Total 1.00 1.00 The total amount of ferrochrome that is available this month to make GPS and HM is 1,050 tonnes. The amount of nickel that is available is 810 tonnes. There is a constraint on production capacity which means that the total combined tonnage of GPS and HM that can be produced this month is not more than 3,600 tonnes. This month Jordao can sell the GPS mix for 7,460/tonne and the HM mix for 9,830/tonne. (i) Formulate a linear program to determine how much of each of the two mixes to make this month so as to maximise profit (define any variables you use, and show what any functions or inequalities represent in your formulation). [5 marks] (ii) Solve the problem in (i) graphically. Support any numerical results with calculations. [8 marks] (iii) (a) Adjust your result in (ii) on the assumption that capacity increases from 3,600 tonnes of the mixes to 3,601 tonnes this month. [4 marks] (b) Hence, find the shadow price of additional production capacity. [2 marks] (c) Calculate the maximum amount to which production capacity can be increased with the same shadow price. [3 marks] (d) - Question continues - Page 3 of 7 Show that the new profit for the capacity calculated in (c) corresponds to the shadow price calculated in (b). [3 marks]