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Cork Institute of Technology
Bachelor of Engineering (Honours) in Structural Engineering – Award
(Bachelor of Engineering in Structural Engineering - Award)
(NFQ – Level 8)
Summer 2005
Systems Analysis
(Time: 3 Hours)
Instructions: Answer FOUR questions. All questions carry equal marks.
Examiners: Mr. D. O’Hare Mr. T. Corcoran Prof. P. O’Donoghue
- (a) (i) What should be the characteristics of the current basis in a linear programming problem in order that the dual simplex method be applicable? How are these characteristics identified in a Simplex table? (ii) Use the dual simplex method to show that the following problem is infeasible:
subject to 2
minimise 2
1 2
1 2
1 2
1 2
x x
x x
x x
z x x
(b) The following table shows a basic solution to a linear programming problem which is feasible but not optimal. Proceed to a solution which is both feasible and optimal and state that solution clearly.
Basis z x 1 x 2 S 1 S 2 S 3 Solution x 2 0 0 1 -1/2 0 0 5 S 2 0 2 0 3/2 1 0 45 S 3 0 1 0 1/2 0 1 35 ( z ) 1 -3 (^0) -5/4 0 0 15/
(c) (i) How many decision variables and how many constraints are there in the dual of the problem in part (b)? What are the optimal values of the dual variables? (ii) What is meant by complementary slackness in the context of linear programming? Support your answer by providing an illustration from the ‘current’ example.
- (a) Solve the following problem using the M-technique or the two-phase method:
subject to 2 3 4 12
minimise 2 9 4
1 2 3
1 2 3
1 2 3
1 2 3
x x x
x x x
x x x
z x x x
(b) The table shown below relates to the following LP problem:
subject to 10
maximise 3 2 2
1 2 3
1 2 3
1 2 3
1 2 3
1 2 3
x x x
x x x
x x x
x x x
z x x x
Basis z x 1 x 2 x 3 S 1 S 2 S 2 Soln. S 1 0 0 0 1/4 1 -1/4 -1/4 A x 1 (^0 1 0) 5/4 0 3/4 -1/4 B x 2 0 0 1 -1/2 0 -1/2 1/2 5/ (z) (^1 0 0) 3/4 0 D 1/4 C
(i) Using the techniques of sensitivity analysis, show that the given basis is optimal, and determine the optimal values of the basic variables and the objective function. (ii) Carry out sensitivity analysis on the objective function coefficients of x 2 and x (^3) (separately) and state your conclusions. (iii) x 3 relates to the production level of product X. If two units of product X are required for a special customer, analyse the impact on the current optimal product mix, and find the revised solution. (iv) A fourth constraint must now also be taken into consideration. That constraint is x 1 (^) + 2 x 2 + x 3 ≤ 10. Does this affect the current solution? Find a new solution, if necessary.
- (c) Four project leaders are available for assignment to three clients. The estimated project completion time (in weeks) for each leader-client combination is shown in the following table:
Client Leader 1 2 3 A 10 15 9 B 9 18 5 C 6 14 4 D 8 18 7 Each client is to be assigned one project leader; it is not acceptable to assign any individual leader to more than one client. Determine the optimum assignment of leaders to clients if the objective is to minimise the total project completion time.
- An investor is considering two investment projects A and B. Both involve outlays of
€1million. Project A will provide a single incoming cash payment after 6 years of €1.7 million. Project B will provide incoming cash payments of €1 million after 6 years, €0. million after 7 years, €0.229 million after 8 years, and €0.245 million after 9 years. (i) Determine the rate of interest at which the net present value of the two projects will be equal. (ii) If the rate of interest is higher than that determined in part (i), which of the two projects will have the greater NPV? Justify your answer. (iii) Determine the internal rate of return for each of the projects.
- (a) Consider the payoff table shown below, along with the information that P(E 1 )=0.3. Payoff
values are in thousands of euro.
Event E 1 Event E 2
Action A 1 50 -
Action A 2 20 10
(i) Calculate the expected monetary value for each action, and hence identify the optimal action according to the expected monetary value criterion (ii) Produce the associated opportunity loss table and determine the best action according to the expected opportunity loss criterion. (b) An oil company is considering an offshore drilling venture. A preliminary investigation indicates that rock formations off the coast are generally favourable to oil and gas deposits. From past information on similar rock formations, the exploration department comes up with the following estimates, where L denotes large deposits and S denotes small deposits: P(L)=0.1, P(S)=0.9. The firm has two possible actions available, to explore or not to explore. Explorations will cost €1.2m. If large deposits are found, the company expects to make a profit of €20m. If the deposits found are small, then the project will be abandoned. What is the expected value of perfect information here?
(c) The company in (b) decides to obtain some additional information. A mining expert would charge €200000 for a preliminary survey and would report the survey results as N, no large deposits likely or Y, large deposits likely. The expert’s surveys are not always correct. The table below is a record of his past performance.
Prediction
Actual result N Y
L 30% 70%
S 85% 15%
Set up a decision tree and determine the optimal course of action for the firm.
