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Cork Institute of Technology
Bachelor of Engineering (Honours) in Structural Engineering- Award
(NFQ Level 8)
Summer 2007
Systems Analysis
(Time: 3 Hours)
Instructions: Answer FOUR questions. All questions carry equal marks.
Examiners: Mr. D. O’Hare Mr. P. Anthony Prof. P. O’Donoghue
- (a) Explain the terms feasible region, slack variable and sensitivity analysis as used in the context of linear programming. (5 marks)
(b) Use the two-phase method to find the optimal solution to the following problem, if it exists: Minimise subject to 2x (^1)
z x x x x x x x
1 2 2 1 2 1 ,^2. (8 marks) (c) Write down the dual of the following problem 1 2 3 1 2 3 1 2 3 1 2 3
Minimise 60 20 75 subject to 10 5 10 275 5 5 10 250 , , 0.
z x x x x x x x x x x x x
(ii) Solve the dual problem graphically. (iii) Using the Complementary Slackness theorem, deduce the solution to the primal problem. (12 marks)
- (a) Use the Dual Simplex method to solve the following problem.
1 2 1 2 1 2 1 2
Minimise 2 subject to 4x 2 6 2 3 1 , , 0
z x x x x x x x
(8 marks)
(b) A company plans production on three of its products - A, B and C. The unit profits on these products are €20, €30 and €20 respectively and they each require two resources, labour and raw material. The following LP problem has been formulated to determine the optimal product mix: 1 2 3 1 2 3
1 2 3
1 2 3
maximise 2 3 subject to 1 (labour) 3 3 3 (^4 7 3) (material) 3 3 3 , , 0.
z x x x x x x
x x x
x x x
Part of the optimal table is shown below:
Basis z x 1 x 2 x 3 S 1 S 2 Solution x 1 4 - x 2 -1 1
(i) Fill in the missing entries in the table and state clearly the optimal solution.
(ii) What should the unit profit on product C be to justify including it as part of the optimal product mix?
(iii) The right hand side of the labour constraint changes from 1 to 4. What now is the solution to the problem?
(iv) A new constraint associated with administrative services must now be considered. The constraint is x (^) 1 + 2 x (^) 2 + x 3 ≤ 4.
What is the solution to the problem now? (17 marks)
- (a) A bank makes four kinds of loans to its personal customers and these loans yield the following annual interest rates to the bank: First mortgage 14% Second mortgage 20% Home improvement 20% Personal overdraft 10%. The bank has a maximum foreseeable lending capability of €250 million and is further constrained by the policies:
- First mortgages must be at least 55% of all mortgages issued and at least 25% of all loans issued (in € terms)
- Second mortgages cannot exceed 25% of all loans issued (in € terms).
- To avoid public displeasure and the introduction of a new windfall tax the average interest rate on all loans must not exceed 15%.
Formulate the bank's loan problem as an LP problem so as to maximise interest income whilst satisfying the policy limitations. Note: It is not necessary to solve the problem which you formulate. (7 marks)
(b) A local authority is under financial pressure and is considering selling some of the land it owns to a building company. The company offers to pay €150m now or €40m a year for the next five years. (i) If the current interest rate is 8%, which option would you advise? (ii) How would your recommendation be affected by changes in the interest rate?
(6 marks)
4 (c) A large organisation is considering replacing part of its vehicle fleet.
Two suppliers have tendered, and the relevant costs and savings are shown below.
Supplier A Supplier B
Year Cost Savings Cost Savings
0 120000 0 85000 0
1 0 25000 35000 0
2 0 30000 0 60000
3 0 40000 0 45000
4 0 50000 0 35000
5 0 25000 0 35000
The company’s current cost of borrowing is 12%. (i) Which supplier would you recommend on the basis NPV? (ii) Calculate the IRR for each supplier, and comment.
(12 marks)
