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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering – Award
(NFQ – Level 8 )
Autumn 2007
Mathematics
(Time: 3 Hours)
Instructions: Answer FOUR questions. All questions carry equal marks. Statistical tables are available.
Examiners: Mr. D. O’Hare Mr. P Clarke Prof M. Gilchrist
- (a) (i) What is the difference between a non-basic feasible solution and a basic feasible solution to a linear programming problem? (ii) If a problem involves 5 decision variables and three constraints of the ≤ type, how many basic variables and how many nonbasic variables are there in any Simplex table
for this problem? (iii) What is the difference between a slack variable and an artificial variable? (6 marks)
(b) Three products A, B and C each demand some of resources R 1 and R 2. The linear programming problem set up in order to maximise profit is as follows:
subject to 6 2 4 200 ..........
maximise 6 4 2
1 2 3
1 2 3 2
1 2 3 1
1 2 3
x x x
x x x R
x x x R
z x x x
x 1 (^) , x 2 , x 3 represent the number of units of A, B, C, respectively, to be produced. Part of the optimal simplex table is as follows:
Basis z x 1^ x 2^ x^3^ S 1 S 2 Solution 1/4 -1/ -1/4 3/
1.(b) contd
(i) Complete the optimal table above and state the optimal solution. Is this optimal solution unique? If not, find an alternative optimal solution. (ii) For each of the decision variables, find the range of values for the objective function coefficient for which the current basis remains optimal. (iii) What is the shadow price for resource R 1? What does this value mean? (iv) A fourth product, D, is to be considered. It requires 2 units of R 1 and 2 units of R 2 and offers a contribution of 9 per unit to the value of the objective function. Is this product worth producing? If it is, find the new optimal solution. (19 marks)
- (a) Use the simplex method to find the solution to the following linear programming problem. Maximise z = 100 x 1 + 200 x 2 + 150 x 3 subject to
1 2 3
1 2 3
1 2 3
≥
x x x
x x x
x x x
(8 marks) (b) Formulate the dual of the problem in part (a). Deduce the solution to the dual from the primal optimal table and verify this solution by solving the dual problem graphically. (7 marks)
(c) Use the two-phase method to solve the following problem. Maximise z = x 1 + 5 x 2 + 3 x 3 subject to
1 2 3
1 2
1 2 3
≥
x x x
x x
x x x
(10 marks)
- (a) A manufacturing firm has developed a transition matrix containing the probabilities that a particular machine will operate or break down in the following week, given its operating condition in the previous week. The matrix is as follows: P= (^)
0.4 is the probability that if the machine is operating in any given week, it will be operating in the following week. (i) Assuming that the machine is operating in week 1, find the probability that it is operating in week 3 and the probability that it is operating in week 4. (ii) Find the steady state probabilities for this chain and interpret the values you obtain.
(iii) The company is considering a preventative maintenance program that would change the operating probabilities as follows: P= (^)
The machine earns the company €15,000 in profit each week it operates. How much should the company be prepared to pay for the maintenance program? (15 marks)
(b) The following is an ANOVA table for a one-way design:
Source df Sum of squares
Mean Square F
Factor * 12.12 4.04 * Error 23 18.90 * Total * * (i) Fill in the values denoted by * in the ANOVA table. (ii) How many factor levels are involved? (iii) How many observations were taken at each factor level? (iv) What conclusion do you draw from the table and why? (10 marks)
- (a) A process engineer has identified two potential causes of electric motor vibration, the material used for the motor casing and the supply source of bearings used in the manufacture of the motor. The table below shows data gathered on vibration in motors, with casings made of steel, aluminium and plastic, which were constructed using bearings supplied by three randomly selected sources. Supply Source 1 2 3 Steel 9.1, 9.2 12.3, 11.8 9.7, 10. Material Aluminium 11.0, 10.6 11.5, 12.6 9.9, 10. Plastic 9.8, 10.4 13.4, 12.6 8.5, 8.
Part of the ANOVA table, which allows for the possibility of interaction, is given below. (i) Complete the table and say what conclusions may be drawn from it. (ii) Produce an interaction plot and comment. What recommendation would you make on type of casing material and supply source on the basis of your analysis?
Source Sum of squares df Mean Square F Material Supplier Interactio n
Error 1. Total 37. (12 marks)
- (b) An investigation was carried out to investigate the effect of pH and catalyst concentration on product viscosity. Three replicates of a 2 2 design were carried out with the following results. Catalyst concentration 2.4 2. pH 5.5 192, 198, 189 178, 186, 188 5.8 193, 188, 185 197, 204, 202 (i) Estimate the effect of pH. (ii) Estimate the error variance and the variance of an effect estimate.
2 V E ( (^) i ) (^) n .2^ ek 2
