1 MT263: Problems Do not hand your solutions in for feedback this week. 1. (a) Formulate the following as a LPP. A metal parts company has found that it has some unused time on its machines as follows: machine mins per day idle lathe mill grinder 65 100 90 It has decided to use this idle time to manufacture two particular parts, A and B. It has determined that Each part can be sold in quantities of up to 12 dozen per day. Part A can be sold at a profit of 3. Part B can be sold at a profit of 2. Manufacturing part A takes 5 mins on a lathe, 7 mins on a mill, and 4 mins on a grinder. Manufacturing part B takes 3 mins on a lathe, 9 mins on a mill, and 7 mins on a grinder. How many of parts A and B should be made to maximise the profit? (b) Formulate the following as a LPP. A coffee packer blends Brazilian and Colombian coffee to prepare two products: Super and Deluxe brands. Each kilogram of Super coffee contains 0.5kg of Brazilian coffee and 0.5kg of Colombian coffee, whereas each kilogram of Deluxe coffee contains 0.25kg of Brazilian coffee and 0.75kg of Colombian coffee. The packer has 120kg of Brazilian coffee and 160kg of Colombian coffee available. The profit on each kilogram of Super coffee is 20 cents and the profit on each kilogram of Deluxe coffee is 15 cents. The coffee packer wants to maximise his profit. (c) Formulate the following as a LPP. A farmer can buy two animal feeds F1 and F2. These contain three nutrients N1, N2 and N3 which are needed for the animals. The table shows the nutritional content and cost of each type of feed, and also the minimum daily requirement (MDR) of each nutrient for an animal. The farmer wants to buy feeds as cheaply as possible while satisfying the daily requirements of his animals. F1 N1 5 N2 2 N3 1 Pence/kg 40 F2 MDR 1 11 1 8 2 7 30 2. Put the following LPPs in standard form. (a) minimise subject to (b) maximise subject to 8x1- 4x2 3x1 + x2 >7 =- 2 9x1 + @2 x1, X2 ? 0. 8x1 - 4x2 3x1+ 2 <7 9x1+ x2 <- 2 < 0. (c) minimise subject to 8x1 - 4x2 3x1+ x2 <7 9x1+ x2 ?-2 x1, X2 ? 0. 3. Redo Question 2 but put the LPPs in canonical form instead.
4. Write the following LPPs in matrix form maximise subject to x1-22 + 13 maximise subject to 2x1-x2+ 2x3 ? 4 2x1-3x2 + X3 ?-5 and -x1+ X2- 2x3 ?-1 x1, x2, 13 ? 0. 5. Consider the following linear programming problem 5x1+4x2 + 3x3 2x1+3x2 + x3 = 5 4x1+2 + 2x3 = 11 3x1+4x2 + 2x3 = 8 x1, x2, 13 ? 0. Which of the following are solutions? Which are feasible solutions? Maximise z=3x1+2x2 + X3 subject