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PRIFYSGOL CYMRU/UNIVERSITY OF WALES
ABERYSTWYTH
INSTITUTE OF MATHEMATICS AND PHYSICS
SEMESTER 2 EXAMINATIONS, MAY 2008
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.
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Section A
1 You should NOT use the simplex method or any of its variants in this question. No credit will be given for answers that do so. The region G is defined by the constraints X 1 + X 2 ≤^4 –X 1 + X 2 ≤ 1 2X 1 ≤ 7 X 1 , X 2 ≥ 0 (a) Sketch this region. (b) On your diagram identify (i) a basic feasible solution, (ii) a non-basic feasible solution. (c) If the right hand side of the first constraint changes, to what value must a increase before X 1 + X 2 ≤ a becomes redundant? (d) Identify the point at which X 1 + 2X 2 is maximized over G. (e) What is the solution if X 1 and X 2 have to take integer values? (f) If the final constraint is replaced by 2X 1 + bX 2 ≤ 7, what values can b take for the solution to remain unchanged?
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2 Use the simplex method to solve the problem: Maximize P = 6X 1 + 2X 2 + 5X 3 Subject to –4X 1 + 2X 2 + X 3 ≤ 4 2X 1 + X 3 ≤ 20 X 1 – X 2 – X 3 ≤ 10 X 1 , X 2 , X 3 ≥ 0 [8]
3 When is the dual simplex method used? The initial dual simplex tableau for a particular problem is: X 1 X 2 X 3 X 4 X 5 X 6 RHS –1 2 3 1 0 0 – 1 –1 –1 0 1 0 – 2 –2 –2 0 0 1 – –3 –1 –2 0 0 0 –Q (i) What is the objective in this problem? (ii) Perform ONE iteration of the dual simplex algorithm. (iii) What do you conclude?
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Section B
7 A bus company operates buses on several routes between 7.30am and 7.30pm. Timetables are such that a minimum of 50 drivers are needed between 7.30am and 9.30am, at least 30 between 9.30am and 4.00pm, at least 45 between 4.00pm and 6.00pm and at least 20 between 6.00pm and 7.30pm. On any given day a driver will work one of three shifts: Morning shift: 7.30am – 1.30pm Afternoon shift: 1.30pm – 7.30pm Split shift: 7.30am – 9.30am and 4.00pm – 6.00pm For contractual reasons, split shift personnel cannot account for more than one-third of the workforce. (a) If the company decides to employ X 1 morning shift drivers, X 2 afternoon shift drivers and X 3 drivers working a split shift, formulate a linear programming problem to determine the numbers of drivers working each shift so as to minimise the total number of drivers required. (b) Identify any redundant constraints in your formulation. The optimal tableau for this problem is: X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 RHS 1 0 0 0 –1 0 0 0 0 30 0 0 0 2 –3 –1 0 0 1 20 0 0 0 –1 1 –1 1 0 0 5 0 0 1 –1 1 0 0 0 0 20 0 0 0 0 0 –1 0 1 0 10 0 1 0 0 0 –1 0 0 0 30 0 0 0 –1 0 –1 0 0 0 D– (c) What is the solution? (d) How many solutions are there for which the Xi take integer values?
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8 Rose and Colin each have a 2p, a 5p and a 10p coin and each simultaneously display one of her/his coins. If the total value of the coins shown is odd then Colin gives Rose the coin he is displaying; if the
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total is even than Rose gives her displayed coin to Colin. (a) Write down the payoff matrix for this game. (b) If Colin chooses one of his three coins at random (and Rose knows this), what should Rose do? After removing dominated strategies, the following three tableaux derive the solution to Colin’s problem: 0 7 1 0 1 4 -3 0 1 1 1 1 0 0 P 0 7 1 0 1 1 -0.75 0 0.25 0. 0 1.75 0 -0.25 P-0. 0 1 1/7 0 1/ 1 0 3/28 1/4 5/ 0 0 -1/4 -1/4 P–1/ (c) Explain how the problem was formulated. (d) How should Colin play the game? (e) How should Rose play the game? (f) Does the game favour either player?
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9 Andy, Balazs, Clive and Dave are asked to deliver four special lectures (one each) between them. Their departmental head offers a cash sum as an incentive, and invites each to name his price for the four lectures, with the stipulation that the sums should be no more than £900 and that they must quote just £200 for at least one of the lectures. The tenders received (in £100’s) are: Lecture 1 Lecture 2 Lecture 3 Lecture 4 Andy 5 5 9 2 Balazs 7 4 2 3 Clive 9 2 5 9 Dave 6 2 6 7 Use the Hungarian method to allocate the lectures for the least cost.
At the last minute Edwin comes along and offers prices of £500, £450, £200 and £800 for the four lectures respectively. Should the Head of Department accept Edwin instead of one of the other four for one of the lectures?
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