Linear Programming Examination January 2010, Exams of Linear Programming

Download Linear Programming Examination January 2010 and more Exams Linear Programming in PDF only on Docsity! SEFYDLIAD MATHEMATEG A FFISEG INSTITUTE OF MATHEMATICS AND PHYSICS SEMESTER 1 EXAMINATIONS, JANUARY 2010 MA33110 – Linear Programming Time allowed – 2 hours • All questions may be attempted. • Marks gained from questions in section B will be given greater consideration in assessing a first class performance. • Calculators are permitted, provided they are silent, self-powered, without communications facilities, and incapable of holding text or other material that could be used to give a candidate an unfair advantage. They must be made available on request for inspection by invigilators, who are authorised to remove any suspect calculators. MA33110: Linear Programming January 2010 Page 2 of 7 Printed: 04/04/2010 Questions 1-4 are based on the following diagram. The diagram is also printed on a separate sheet appended to this examination paper. You may use that sheet for your answers, in which case you should detach it and insert it in your answer book. Remember to write your University number in the appropriate space. Related linear system –X1 + X2 X3 = 3 X1 +4X2 X4 = 28 8X1 –3X2 X5 = 35 3X1 + X2 X6 = 11 –2X1 +5X2 X7 = 4 MA33110: Linear Programming January 2010 Page 5 of 7 Printed: 04/04/2010 Section B 9 A manufacturer of aluminium cans must produce monthly at least 2400 cases of a Standard can and a minimum of 2800 cases of Toughened cans. The company has three manufacturing processes it can use; the first uses a special pure grade aluminium, whilst the other two allow for some use of recycled aluminium. To comply with the conditions of government funding already received, the company must use at least 600kg of recycled aluminium in its monthly production. The characteristics of each process are shown in the following table: Input (per run) Output (per run) Recycled aluminium used (kg) Standard cans (cases) Toughened cans (cases) Cost (per run) Process 1 0 6 8 65 Process 2 2 12 12 150 Process 3 3 10 15 200 The company manager must decide how many runs of each process to initiate per month to run the plant at minimum cost. With Xi denoting the number of runs of the i-th process per month, for i=1,2,3 the following table shows the initial and final tableaux when the problem was solved using the dual simplex method: X1 X2 X3 X4 X5 X6 RHS Initial tableau 0 –2 –3 1 0 0 –600 –6 –12 –10 0 1 0 –2400 –8 –12 –15 0 0 1 –2800 –65 –150 –200 0 0 0 –Q Final tableau –3/4 0 1 –3/4 1/8 0 150 9/8 1 0 5/8 –15/80 0 75 –23/4 0 0 –15/4 –3/8 1 350 –185/4 0 0 –225/4 –25/8 0 –Q+41250 (a) Explain how the initial tableau was derived and give the solution. (b) How would the solution change if the legal requirement of recycled aluminium usage was reduced to 560 kg per month? (c) If the cost of using Process 3 changes to £(200+d), for what range of values of d does the solution remain valid? (d) How would you interpret the value “–25/8” in the objective row of the final tableau? [5] [5] [5] [3] MA33110: Linear Programming January 2010 Page 6 of 7 Printed: 04/04/2010 10 The following tables show the simplex iterations for solving the problem of maximising 9X1+2X2+2X3 subject to X1+X2–X3≤5 and 2X1–X2+3X3≤8 in non- negative variables X1, X2 and X3. X1 X2 X3 X4 X5 RHS 1 1 –1 1 0 5 2 –1 3 0 1 8 9 2 2 0 0 P 0 3/2 –5/2 1 –1/2 1 1 –1/2 3/2 0 1/2 4 0 13/2 -23/2 0 -9/2 P-36 0 1 –5/3 2/3 –1/3 2/3 1 0 2/3 1/3 1/3 13/3 0 0 –2/3 –13/3 –7/3 P–121/3 Apply Gomory’s method to the first row of the final tableau and modify the solution if the variables have to be integer valued. [10] 11 Write the linear programming problem Minimise 2X1 – X2 subject to 5X1 + 7X2 ≤ 30 X1 – X2 ≥ 2 X1 + 8X2 = 11 X1 , X2 ≥ 0 as a maximisation problem in standard form. Hence write down its dual, expressing it as concisely as possible. Given that the solution to the primal problem occurs at X1=3, X2=1, use the principle of Complementary Slackness to deduce the solution to the dual. Verify that the optimal values of the objective functions in the two problems are related as you would expect. [10] 12 Rose and Colleen each have two cards labelled 1 and 2. Each player must choose and display one card. If the sum of the two numbers displayed is even, Rose wins that amount from Colleen; if, on the other hand, the sum is odd then Colleen wins that amount from Rose. How should Colleen play the game? Does the game favour either player? [12] MA33110: Linear Programming January 2010 Page 7 of 7 Printed: 04/04/2010 13 A supermarket chain has available 25 pallets of a particular product available at each of their three warehouses. They needs to transport 15 pallets to the outlet at Alpha Avenue and 20 pallets to each of the shops at Beta Bay, Gamma Grove and Delta Drive. The following table displays the transportation costs per pallet from each warehouse to each outlet: Outlet Alpha Avenue Beta Bay Gamma Grove Delta Drive W a re h o u se W1 2 8 22 17 W2 5 10 25 18 W3 1 4 19 20 (a) Use the Northwest Corner rule to determine a basic feasible transportation schedule. Verify that the cost of this is £1080. (b) A colleague comes up with the alternative schedule: From W1: 15 to Alpha Avenue, 10 to Gamma Grove From W2: 5 to Beta Bay, 20 to Delta Drive From W3: 15 to Beta Bay, 10 to Gamma Grove which costs just £910. Apply the Method of Fictitious Costs to show that this schedule is in fact optimal. . [3] [7]