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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering-
Award
(EMECH_8_Y4)
Summer 2008
Mathematics
(Time: 3 Hours)
Instructions: Answer FOUR questions. All questions carry equal marks. Statistical tables are available.
Examiners: Mr. D. O’Hare Prof. M. Gilchrist Mr. P Clarke
- (a) Use the Simplex method to solve the following LP problem and then verify your solution by solving the problem graphically.
Maximise subject to
z x x x x x x x x
1 2 1 2 1 2 1 ,^2.
(10 marks)
(b) (i) Use either the dual simplex procedure or the two-phase method to find the solution to the following problem, if it exists: 1 2 1 2 1 2 1 2
Minimise 2 5 subject to 2x 4 3 6 , 0.
z x x x x x x x
(ii) Write down the dual of the problem in part (i), and deduce its solution from the final table above. Give two illustrations of the complementary slackness theorem as it applies in this example. (15 marks)
- Consider the following linear programming problem, along with part of the associated optimal table below. 1 2 3 1 2 3 1 2 3 1 2 3
maximise 3 7 2 subject to 100 3 200 , , 0.
z x x x x x x x x x x x x
Basis z x 1 x 2 x 3 S 1 S 2 Solution x 1 3/2 -1/ x 2 -1/2 1/
(i) Using the techniques of sensitivity analysis , fill in the missing entries in the table and state clearly the solution to the problem. (6 marks) (ii) Within what limits should the objective function coefficient of (^) x 2 lie in order that the current basis remains optimal? What changes in basis will occur if the coefficient lies outside this interval? (5 marks) (iii) What are the shadow prices of the two constraints here, and what do these values mean? (2 marks) (iv) If a new constraint, x 1 (^) + x 2 (^) + x 3 ≤ 90 , is added to the problem, is the current basis still feasible? If not, find a new basis that is both feasible and optimal. (6 marks) (v) A fourth variable, x (^) 4 , is introduced so that the problem now reads: 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3
maximise 3 7 5 6 subject to 100 3 2 200 , , 0.
z x x x x x x x x x x x x x x x
Find the solution to the revised problem. (6 marks)
- (a) Four components, A-D, can be used to manufacture a paint, where the following specifications apply: (i) boiling point should be at least 70°C (ii) hardening time should not exceed 4 hours; (iii) at most 70% of volume should be plastic; (iv) at least 5% of volume should be acid. The contributions of each of the components to the above characteristics are in proportion to its part in the mixture. The relevant data are in the following table:
A B C D
Boiling point 60 99 15 30 Hardening time(h) 1 4 3 2 % plastic 80 50 2 14 % acid 20 0 0 10 Cost( €/litre ) 3.0 7.5 4.5 2.
- (a) The following table gives data for a 2X3 factorial experiment with two observations for each factor-level combination.
Factor B Level 1 2 3 Factor A 1 3.1, 4.0 4.6, 4.2 6.4, 7. 2 5.9, 5.3 2.9, 2.2 3.3, 2. (i) Draw an interaction plot for this set of data, and comment. (ii) Given that the interaction and the total sum of squares are 18.007 and 28. respectively, compile the ANOVA table for this experiment. What conclusions do you draw from the F-ratios involved? (12 marks) (b) A study is conducted on the effect of temperature, time in process, and rate of temperature rise on the amount of dye (in mg) left in the residue bath in a dyeing process. The experiment was run at two levels of temperature ( 120^0 C and 140 0 C ), two levels of time in the process ( 40 min, 55 min ) and two rates of temperature rise ( R 1 and R 2 ). Two readings were taken at each combination of factor levels. The resulting set of data is as follows:
Temperature 120 0 140 0 40 min 55 min 40 min 55 min Rate R 1 19.9, 18.6 17.4, 16.8 25.0, 22.8 19.5, 18. R 2 14.5, 16.1 16.3, 14.6 27.7, 18.0 28.3, 26. (i) Estimate the error variance here. (ii) Calculate the temperature contrast, the temperature-time interaction contrast, and estimates of the associated effects. (iii) Test the significance of the effect estimates obtained in part (ii) using both a t- test and an F-test. (13 marks)
- It is assumed that the batteries for a set of 100 lamps are replaced at the end of the week in which they fail. 20% of batteries fail during their first week of operation, 50% during their second and 30% during their third week. You are required to model this situation as a Markov chain. (i) Identify three states and produce the state transition diagram for the chain. (8 marks) (ii) Write down the transition matrix, and predict the number of replacement batteries required at the end of each of the first three weeks. Assume that at the beginning of the first week, all batteries are new. (9 marks) (iii) Find the number of replacement batteries required at the end of each week, in the long run. (8 marks)
ANOVA
1. One-way model : y ij =μ+τ i +ε ij , i = 1 , 2 ,..., a , j = 1 , 2 ,...., n.
Total SS: ∑∑ −
an
y y ij
2 ..^2
Factor SS: an
y n
a y i
i^2 1
=
2. Randomised block model : y ij =μ+τ i +β j +ε ij , i = 1 , 2 ,..., a , j = 1 , 2 ,...., b.
Total SS: ∑∑ −
ab y y ij
2 ..^2
Factor SS: ab
y b
a y i
i^2 1
=
Block SS: ab
y a
b y j
j^2 1
.^2
=
