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CORK INSTITUTE OF TECHNOLOGY INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Autumn Examinations 2008/
Module Title: Mathematics for Engineering 402
Module Code: STAT
School: Mechanical and Process Engineering
Programme Title: Bachelor of Engineering (Honours) in Mechanical Engineering - Award
Programme Code: EMECH_8_Y
External Examiner(s): Dr Paul Robinson Internal Examiner(s): Mr. D. O’Hare
Instructions: Answer any three questions. All questions carry equal marks.
Duration: 2 HOURS
Sitting: Autumn 2009
Requirements for this examination: Statistical tables by Murdoch and Barnes.
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the correct examination paper. If in doubt please contact an Invigilator.
Q1. (a) A manufacturing process involves two stages, A and B. There are 4500 spare man-machine hours available in stage A and 4000 spare man-machine hours available in stage B. Four products are to be considered for manufacture and the relevant time requirements are shown below: Time requirement, in hours, per 100 units Product 1 2 3 4 Stage A 10 30 80 40 Stage B 20 10 10 30 The profit levels for the products, per unit, are €10, €10, €40, and €30, respectively. Formulate the above problem as a linear programming problem where the objective is to maximise total financial return. Find the optimal solution using the simplex method, and state clearly what this solution is. [13 marks]
(b) Use the two-phase method to solve the following problem:
maximise subject to
z x x x x x x x x
1 2 1 2 1 2 1 ,^2. [13 marks]
(c) Verify the solution obtained in part (b) by solving the problem graphically. [7 marks]
Q2. (a) Consider 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
- (a) A personnel manager has recently hired 25 workers, each of whom will work in either assembly, machining, packing or inspection. 15 of the workers are classified as apprentices while the remaining 10 are classified as experienced. The company has developed productivity measures for the workers as follows:
Productivity per worker by position Assembly Machining Packing Inspection Apprentice 58 62 76 29 Experienced 73 94 81 56
Seven workers each are required in assembly, machining, and packing, while four are needed in inspection. Treat the problem of allocating workers to departments as a transportation problem, and thus find the allocation that will maximise the overall sum of productivity measures. [10 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 24 18.90 * Total * * (i) Fill in the values denoted by * in the ANOVA table. (ii) How many factor levels are involved? How many observations were made overall? (iii) What conclusion do you draw from the table and why? (10 marks)
(c) 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 is given below. Use the formula given to calculate the material sum of squares. Complete the table and say what conclusions may be drawn from it.
Source Sum of squares df Mean Square F Material Supplier Interaction 6.5256 4 Error 1. Total 37.
(13 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: ∑∑ y ij − aby
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
=
- AxB factorial design model :
y ijk =μ+τ i +β j +(τβ ) ij +ε ijk , i = 1 , 2 ,..., a , j = 1 , 2 ,...., b , k = 1 , 2 ,..., n.
SST: abn
y y ijk
2 ...^2
SSA: abn
y bn
a y i
i^2 1
SSB: abn
y an
b y j
j^2 1
.^2.
=
2 k^ design, n replicates.
Effect estimate given by (^1)
. 2
n k −
Contrast
Effect SS given by (^) k n
Contrast
. 2
( )^2
