Download Examination Paper: Mathematics for Engineering 402, LP Problems and Optimization and more Exams Engineering Mathematics in PDF only on Docsity! CORK INSTITUTE OF TECHNOLOGY INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ Semester 1 Examinations 2009/2010 Module Title: Mathematics for Engineering 402 Module Code: STAT8002 School: Science Programme Title: Bachelor of Engineering (Honours) in Mechanical Engineering - Award Bachelor of Engineering (Honours) in Biomedical Engineering - Award Programme Code: EMECH_8_Y4 EBIOM_8_Y4 External Examiner(s): Mr. J. Reilly Internal Examiner(s): Mr. D. O’Hare Instructions: Answer any three questions. All questions carry equal marks. Duration: 2 HOURS Sitting: Winter 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. 2 1. (a) A company manufactures three models of a certain product. Each model has to go through three different operations: moulding, assembly, and finishing. The table below gives all the relevant information. Model Moulding (hrs/unit) Assembly (hrs/unit) Finishing (hrs/unit) Profit (€/unit) 1 3 6 10 160 2 2 3 8 120 3 4 6 12 240 Capacity/wk 48 hr 72 hr 160 hr Formulate the problem of finding the product-mix which will maximise weekly profit as an LP problem. Set up the initial Simplex table, perform one iteration of the Simplex method, and comment. (13 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) Verify the solution obtained in part (i) by solving the problem graphically. (iii) 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. (20 marks) 2. Consider the following linear programming problem, along with the associated optimal table below (with some entries missing). 1 2 3 1 2 3 1 2 3 1 2 3 maximise 3 7 5 subject to 100 2 3 200 , , 0. z x x x x x x x x x x x x = + + + + ≤ + + ≤ ≥ 5 4. (a) A function ( )f x is of period 2π and is defined over a cycle by 0 ( ) - 2 x if x f x x if x π π π π π − ≤ ≤ = ≤ ≤ Find a Fourier Series for this function. ( ) ( ) ( ) ( ) ( ) ( ) 2 2 ( - )cos sin Note: ( - )sin - ( - )sin cos ( - )cos x nx nx x nx dx n n x nx nx x nx dx n n π π π π = − = − ∫ ∫ (13 marks) (b) A uniform rod is aligned along the x-axis between the points x=0 and x=L. Both ends are maintained at a temperature of 200C. The temperature u(x,t) at any point at any instant is found by solving the partial differential equation 2 2 x uk t u ∂ ∂ = ∂ ∂ The initial temperature distribution is given by u(x,0)=f(x). By using a substitution v(x,t)=u(x,t)-20 solve this partial differential equation. In particular find the solution where 4( ) 20 xf x L = + . Note: 2 2 2 sin cos sinn x Lx n x L n xx dx L n L n L π π π π π = − + ∫ 2 2 2 cos sin cosn x Lx n x L n xx dx L n L n L π π π π π = + ∫ (20 marks) 6 ANOVA 1. One-way model: .,....,2,1,...,,2,1, njaiy ijiij ==++= ετµ Total SS: ∑∑ − an y yij 2 2 .. Factor SS: an y n ya i i 2 1 2 . ..−∑ = 2.Randomised block model: .,....,2,1,...,,2,1, bjaiy ijjiij ==+++= εβτµ Total SS: ∑∑ − ab y yij 2 2 .. Factor SS: ab y b ya i i 2 1 2 . ..−∑ = Block SS: ab y a yb j j 2 1 2 . ..−∑ = 3. AxB factorial design model: .,...,2,1,,....,2,1,...,,2,1,)( nkbjaiy ijkijjiijk ===++++= ετββτµ SST: abn y yijk 2 2 ...−∑∑∑ SSA: abn y bn ya i i 2 1 2 .. ...∑ = − SSB: abn y an yb j j 2 1 2 .. ...∑ = − 2k design, n replicates. Effect estimate given by 12. )( −kn Contrast Effect SS given by kn Contrast 2. )( 2
