Download University of Wales, Aberystwyth - Linear Programming Exam Questions, May/June 2009 and more Exams Linear Programming in PDF only on Docsity!
PRIFYSGOL CYMRU / UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 2 EXAMINATIONS, MAY/JUNE 2009
MA33110 โ Linear Programming
Time allowed โ 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.
Graph paper will be provided.
Section A
Questions 1-4 refer to the region K described and partly drawn on this page. An additional copy of this page is provided at the end for you to use in answering these questions. Remember to detach that answer sheet and insert it in your answer book.
Region K
Constraint Associated slack variable (1) โX 1 + 3X 2 โค 30 X 3 (2) X 1 + 3X 2 โฅ 18 X 4 (3) X 1 + 2X 2 โค 30 X 5 (4) 3X 1 โ X 2 โค 24 X 6 X 1 , X 2 โฅ 0
0
5
10
15
20
0 5 10 15 20
5 When and why are artificial variables, as opposed to slack variables, used? Use an artificial variable method of your choice to solve the problem: Minimise X 1 + 2X 2 + 3X 3 Subject to X 1 + X 2 + X 3 โฅ 5
- X 1 + X 2 โ 3X 3 โค 4 X 1 , X 2 , X 3 โฅ 0
[2]
[7]
6 When maximising P = 12X 1 + 7X 2 + 4X 3 subject to certain constraints on non-negative variables X 1 , X 2 , X 3 , the following simplex tableau was obtained: X 1 X 2 X 3 X 4 X 5 X 6 RHS 8/5 0 0 1 3/5 โ4 136/ 8/5 0 1 0 3/5 โ2 56/ 4/5 1 0 0 โ1/5 3/4 8/ 0 0 0 0 โ1 โ1 Pโ Find all solutions to the problem. [9]
7 Explain the terms^ payoff matrix^ and^ dominance^ in the context of two person zero sum games. Solve the game with payoff matrix 0 1 5 7 5 10 15 4 5 5 0 10 5 10 0
[4]
[6]
Section B
8 Aber Wines are to produce three types of wine: a sweet wine, a regular and a dry. They have available 200kg of Great Grapes, 168 kg of Dodgy Grapes, 50kg of sugar and 244 labour-hours. The raw materials, labour requirements and profit for every 10 litres of these wines are shown in the following table: Great Grapes (kg)
Dodgy Grapes (kg)
Sugar (kg)
Labour (hrs) Profit (ยฃ) Sweet 6 6 5 12 90 Regular 12 0 2 10 70 Dry 0 12 0 6 120 The company want to know the product mix of wines that will maximise their profits. (a) Formulate this as a linear programming problem using X 1 , X 2 and X 3 to represent the amounts of Sweet, Medium and Dry respectively (in units of 10 litres). Using X 4 , X 5 , X 6 and X 7 for the slacks associated with the resources, in the order of the columns of the table above, construct the initial simplex tableau. The final tableau is as follows: X 1 X 2 X 3 X 4 X 5 X 6 X 7 RHS โ24/5 0 0 1 3/5 0 โ6/5 8 1/2 0 1 0 1/12 0 0 14 16/5 0 0 0 1/10 1 -1/5 18 9/10 1 0 0 โ1/20 0 1/10 16 โ33 0 0 0 โ13/2 0 โ7 Pโ (b) Give the optimal solution, and interpret the entries 18 and โ7 in this table. (c) In terms of profit per litre, when would it be viable to produce the Sweet wine? (d) The company is considering ordering 50 kg more grapes of one type. Which type would you recommend them to buy, and what should they be prepared to pay for them? If they follow your advice and buy 50kg of the grapes you advocate at ยฃ5 per kg, what is the increase in profit?
[4]
[6]
[2]
[6]
6 4 5 8 3 7 8 7 11 9 2 7 8 6 6 10 3 7 2 7 3 3 4 4 4 3 4 5 2 9 9 3 4 7 2 8 3 1 2 5 0 4 6 5 9 7 0 5 5 3 3 7 0 4 0 5 1 1 2 2 2 1 2 3 0 7 7 1 2 5 0 6 3 0 1 4 0 2 6 4 8 6 0 3 5 2 2 6 0 2 0 4 0 0 2 0 2 0 1 2 0 5 7 0 1 4 0 4 (a) Explain what she has done here. (b) Complete the solution. (c) Explain (without carrying out any calculations) in what ways she would have accommodated: (i) a stipulation that Brian was not prepared to go to Erwgoch; (ii) the availability of a seventh taxi driven by George.
[2]
[8]
[4]
11 Colin and Rowland are the only two candidates in a local election. The constituency covers three wards and polling has shown that all three are undecided over the two candidates. The two candidates have to decide which wards to focus their campaigns on during the last few weeks of their campaign. Opinion polls also suggest that if both candidates concentrated their campaigns on the East ward Rowland would lose by 200 votes; if both concentrated on the North ward however Rowland would win by 500. Similar assessment for the other combinations of where the candidates concentrate their campaigns are shown in the following table, the entries representing the net gain to Rowland:
Colin goes to East North South
Rowlandgoes to
East^ โ200^ โ300^ + North โ500 +500 โ South +300 0 0 (a) How do you know that this payoff matrix favours Rowland? The problem of finding Rowlandโs best (mixed) strategy is formulated as a linear programming problem, and the initial and final tableaux are shown below: X 1 X 2 X 3 X 4 X 5 X 6 RHS
INITIALTABLEAU
2 5 โ3 1 0 0 โ 3 โ5 0 0 1 0 โ โ3 1 0 0 0 1 โ โ1 โ1 โ1 0 0 0 โP
FINALTABLEAU
0 0 1 โ1/3^ 17/36^ โ25/36^ 3/ 0 1 0 0 1/4 โ1/4 1/ 1 0 0 0 โ1/12 5/12 1/ 0 0 0 โ1/3 โ29/36 โ49/36 โP+5/
(b) Explain how the initial tableau was formed. (c) What should Rowland do? (d) Which ward should Colin visit most often? (e) If both candidates follow their optimal strategies, who wins the election, and what would you expect the winning majority to be?
[2]
[5]
[3]
[3]
[3]
9
0
5
10
15
20
0 5 10 15 20
