ASSIGNMENT BOOKLET. M.Sc. (Mathematics with Applications in Computer Science) Mathematical Modelling (January 2014 November 2014)

Transcription

1 ASSIGNMENT BOOKLET MMT-009 M.Sc. (Mahemaics wih Applicaions in Compuer Science) Mahemaical Modelling (January 014 November 014) School of Sciences Indira Gandhi Naional Open Universiy Maidan Garhi New Delhi

2 Dear Suden, Please read he secion on assignmens and evaluaion in he Programme Guide for Elecive courses ha we sen you afer your enrolmen. A weighage of 0 per cen, as you are aware, has been assigned for coninuous evaluaion of his course, which would consis of one uor-marked assignmen. The assignmen is in his bookle. Insrucions for Formaing Your Assignmens Before aemping he assignmen please read he following insrucions carefully. 1) On op of he firs page of your answer shee, please wrie he deails exacly in he following forma: ROLL NO.: NAME : ADDRESS : COURSE CODE :. COURSE TITLE :. ASSIGNMENT NO.:. STUDY CENTRE :.... DATE :.... PLEASE FOLLOW THE ABOVE FORMAT STRICTLY TO FACILITATE EVALUATION AND TO AVOID DELAY. ) Use only foolscap size wriing paper (bu no of very hin variey) for wriing your answers. 3) Leave 4 cm margin on he lef, op and boom of your answer shee. 4) Your answers should be precise. 5) While solving problems, clearly indicae which par of which quesion is being solved. 6) This assignmen is o be submied o he sudy cenre. You canno fill he exam form for his course ill you have submied his assignmen. So solve i and submi i o your sudy cenre a he earlies. We srongly sugges ha you reain a copy of your answer shees. 7) This assignmen is valid only upo November, 014. If you have failed in his assignmen or fail o submi i by November 014, hen you need o ge he assignmen for he year 015. We wish you good luck.

3 Assignmen (MMT-009) (January 014 November 014) Course Code: MMT-009 Assignmen Code: MMT-009/TMA/014 Maximum Marks: a) A company manufacuring sof drinks is hinking of expanding is plan capaciy so as o mee fuure demand. The monhly sale for he pas 5 years are available. Sae, giving reasons, he ype of modelling you will use o obain good esimaes for fuure demand so as o help he company make he righ decisions. Also sae four essenials and four non-essenials for he problem. (6) b) Which one of he following porfolios canno lie on he efficien fronier as described by Markowiz? Porfolio Expeced reurn Sandard deviaion W 9% 1% X 5% 7% Y 15% % Z 1% 15% (3). a) Le G () be he amoun of he glucose in he bloodsream of a paien a ime. Assume ha he glucose is infused ino he bloodsream a a consan rae of k g / min. A he same ime, he glucose is convered and removed from he bloodsream a a rae proporional o he amoun of he glucose presen. If a = 0, G = G(0) hen i) formulae he model. ii) find G () a any ime. iii) discuss he long erm behavior of he model. (3) 9 b) Tumour is developing from he organ of a human body wih concenraion wih growh and decay conrol parameers 9. and.7 respecively. In how many days he size of he umour will be wice is size? (4) 3. a) Reurn disribuions of he wo securiies are given below: Reurn Probabiliies X Y p xj = p yj = p j Find which securiy is more risky in he Markowiz sense. Also find he correlaion coefficien of securiies X and Y. (8) 3

4 b) Le P = ( w 1, w ) be a porfolio of wo securiies X and Y. Find he values of w 1 and w in he following siuaions: i) ρ = 1 and P is risk free. ii) xy σ = σ and variance P is minimum. x y iii) Variance P is minimum and ρ 0.5, σ = and σ = 3. (6) xy = x 4. a) Companies considering he purchase of a compuer mus firs assess heir fuure needs in order o deermine he proper equipmen. A compuer scienis colleced daa from seven similar company sies so ha compuer hardware requiremens for invenory managemen could be developed. The daa colleced is as follows: y Cusomer Orders (in housands) Add-delee iems (in housands) CPU ime (in hours) i) Find a linear regression equaion ha bes fi he daa. (4) ii) Esimae he error variance for he regression model obained in i) above. () b) The populaion consising of all married couples is colleced. The daa showing he age of 1 married couples is as follows: Husband s age Wife s age Husband s age Wife s age i) Draw a scaer plo of he daa (1) ii) Wrie wo imporan characerisics of he daa ha emerge from he scaer plo. () iii) Fi a linear regression model o he daa and inerpre he resul in erms of he comparaive change in he age of husband and wife. (3) iv) Calculae he sandard error of regression and he coefficien of deerminaion for he daa. (3) 5. Consider a discree model given by r N N = = ( ), > + f N r b N. 1 4

5 Invesigae he linear sabiliy abou he posiive seady sae * N by seing N = N * + n. Show ha n saisfies he equaion 1 n + 1 n + ( r 1) r n 1 = 0. Hence show ha r = is a bifurcaion value and ha as r he seady sae bifurcaes o a periodic soluion of period 6. (10) 6. a) The populaion dynamics of a species is governed by he discree model x = n xn+ 1 xn exp r 1, K where r and k are posiive consans. Deermine he seady saes and discuss he sabiliy of he model. Find he value of r a which firs bifurcaion occurs. Describe qualiaively he behaviours of he populaion for r = + ε, where 0 < ε << 1. Since a species becomes exinc if x n 1 for any n > 1, show using ieraions, ha irrespecive of he size of r > 1 he species could become exinc if he carrying capaciy r 1 k < r exp[1 + e r]. (7) b) Do he sabiliy analysis of he following model formulaed o sudy he effec of oxican on prey-predaor populaion and inerpre he soluion. dn 1 = r0 N1 r N 1C0 N1 bn1 dn = d0n d N 1V 0N + β0bn1 dc 0 = k1p g C 1C0 m1 0 dv 0 = kp g V V0 m 0 dp = Q hp kp ( N1 + N ) + gc0n1 +lv0n. Where all he variables and consans are same as defined in he sysem (3)-(35) of Block- of MMT-009 course excep for he following Q = consan inpu rae h = decay rae P() = environmenal oxican concenraion k = ingesion rae of oxican by he populaions g, l = reurn rae of oxican in he environmen afer he deah of he populaions, assuming ha oxican is non-degradable Q > 0, h, k, g, l are posiive consans. (8) 7. Consider he advecion-diffusion-reacion model of a single species populaion given by he following equaion: N N 1 N N = rn 1 + D v, 0 x K x x The iniial and boundary condiions are as follows: N ( x, 0) = f ( x) > 0 x; 0 x 5

6 N = a x = 0 and x =, i = 1, * N i * where N i are he seady-sae soluions of he given model. Obained he seady-sae soluions and do heir sabiliy analysis. Inerpre he soluions obained and also wrie he limiaions of he model. (10) 8. a) Maximize Z = 3x1 + x + 3x3, subjec o he consrains x 1 + x + x3 4, 4x 3x3, x1 3x + x3 3, ( x 1, x, x3) 0 and are inegers. (8) b) A ax consuling firm has 4 service couners in is office for receiving people who have problems and complains abou heir income, wealh and sales axes. Arrival average 80 persons in an 8-hour service day. Each ax advisor spends an irregular amoun of ime servicing he arrivals, which have been found o have an exponenial disribuion. The average service ime is 0 minues. Calculae he average number of cusomers in he sysem, average number of cusomers waiing o be serviced, average ime a cusomer spends in he sysem and average waiing ime for a cusomer. Calculae how many hours each week does a ax adviser spend performing his job. Wha is he probabiliy ha a cusomer has o wai before he ges service? Wha is he expeced number of idle ax advisers a any specified ime? (7) c) In machine mainenance firm, a mechanic repairs four machines. The mean ime beween service requiremen is 5 hours for each machine and forms an exponenial disribuion. The mean repair machine down ime coss Rs.5 per hour and he machine coss Rs.55 per day of an 8 hour day. i) Find he expeced number of operaing machines. ii) iii) Deermine he expeced down ime cos per day. Would i be economical o engage wo mechanics, each repairing only wo machines? (5) 6