O ELECTIVE COURSE : MATHEMATICS O MTE-12 : LINEAR PROGRAMMING

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1 I MTE-12 I BACHELOR'S DEGREE PROGRAMME Term-End Examination December, 2009 O tsr) O ELECTIVE COURSE : MATHEMATICS O MTE-12 : LINEAR PROGRAMMING Time : 2 hours Maximum Marks : 50 Note : Question no. 7 is compulsory. Attempt any four questions from question no. 1 to 6. Calculators are not allowed. 1. (a) Obtain all the basic solutions of the 5 following system of linear equations : x1+2x2+x3-=-4 2x 1 + x2 + 5x3=5. x1, x2-0, which of the solutions are basic feasible? Justify your answer. (b) Solve the following LPP by graphical 5 method. Minimise 2x 1 +3x2 subject to xi +x254 x1+5x2?...4 3x1+x2?-.4 x2.- 0 MTE-12 1 P.T.O.

2 2. (a) Write the dual of the following LPP. 5 max 2x1 + 5x2 + 6x3 subject to 5x1 + 6x2 x3 3 2x1 +x2+4x3=4 x1-5x2 ±3x351 x1, x2, x Your answer should contain at least one unrestricted variable. (b) Solve the following game after reducing it 5 to 2 X 2 game : Player A Player B (a) Use two-phase method to find a basic 5 feasible solution for the LPP. Minimise 5x 1 + 2x2 + 3x3 subject to 3x1 + 4x2 x3?-10 x 1 2x2 + x3?... 5 X1, x2, X3 0 (b) A departmental head has four subordinates, 5 and four are tasks to be performed. The subordinates differ in efficiency, and the tasks differ in their intrinsic difficulty. Her estimate of the time each person would like to perform each task is given in the matrix form as follows : MTE-12 2

3 Tasks Workers E F G H A B \ D CD O How should the tasks be allocated, so as to minimize the total time taken? 4. (a) Following is an initial basic feasible solution 7 for a given balanced transportation problem. 1Q Check whether the solution is optimal. If it is not optimal, carry out as many iterations of the transportation algorithm as necessary and find an optimal solution. (b) Sketch the region { (x, y) I x 2 + y2 31, y2 <_ 3 Is the region convex? Justify your answer. MTE-12 3 P.T.O.

4 5. (a) A manufacturer produces 2 products P 1 and 5 P2. Each unit of P 1 requires 4 hours of grinding, 2 hours of polishing and 5 hours of painting, whereas each unit of P 2 requires 2 hours of grinding, 5 hours of polishing and 3 hours of painting. The manufacturer has 2 grinders, 3 polishers and 2 painters. Each grinder works for 40 hours per week, each polisher works for 60 hours a week and each painter works for 50 hours a week. Unit profit on P 1 and P2 are Rs. 3 and Rs. 4 respectively. Formulate the problem of maximising the profit per week as an LPP. (b) Solve the following game graphically and 5 find the value of the game and the optimal strategies for both the players. Player B 2 5 Player A _ 6. (a) Formulate the following game as LPP models 5 for player A and player B : Player A Player B MTE-12 4

5 Find the initial basic feasible solution of the 3 following transportation problem by North West Corner method. S2 S3 D 1 D2 D ai bj Let A =, = r2 1 5 B 5 3 and C = [ Which of the products ABC, CBA, BAC and BCA are defined? (You need not compute the products.) Justify your answer. MTE-12 5 P.T.O.

6 7. Which of the following statements are true and 10 which are false? Give reasons for your answer. Union of any two convex sets is convex. Both the primal and duel of an LPP can be infeasible. The number of positive allocations in an optimal solution to a transportation problem with m sources and n destinations is m + n 1. The value of the following game is nonnegative : \ ( 0\ 11\ (e) The vectors 1, 1 and 1 are linearly, 0 \0J 1, independent. - o 0 o - MTE-12 6

7 711A R-11 C1 Ch ZEITRT Ch I Ch 4-1.f.1 id TRIATT qk-1 1C4t, 2009.ki-WW: +i 71.Z : Arigeb * 2 gu2 3TAIWUTT 3W : 50 : 177V TY. 7 (4711/ vie1/41 t7 t/ 37: 1 6 # 4 q7) ITT 31.Y1 Of-47/.el?, eta ard7dr ig7 1. (a) - AfINW wii cbtu l f9-wrzi Tit atrtrrft 5 Vrff VI-4R : x1 + 2x2 + x3=4 2.x1 + x2 + 5x3 = 5. x1, X2 0, 14 chlq 3741# 1:ffleTM f? 311:A dtit V.13FEW7R1 (b) 7071 fafq cti-n-r 5 2x1 + 3x2 1 1\1w-14)(u! W.-4R x1 + x25.4 x1 + 5x2?- 4 3x1+x2?-4 x1, x2 0 MTE-12 7 P.T.O.

8 2. (a) F-P-irriFoci LPP tth - F T : 5 2x1 +5x2 +6x3 f aftl-*71:11-*-71:1t 5x1 + 6x2 x3 3 2x1+x2+4x3=4,C1-5x2 +3x3 1 x1, x2, x Trcrk drit wa. -wrf tm rr--4-r (b) 1-1 igcf 2 x 2 tart Tigedff cht 5 ftg ciisl B tiriisl A (a) f? tr u i fa-ry rod LPP13-TRTR-1.-ti*rd 5 5x1 +2x2 +3x3 'W1 : 3x 1 + 4x2 x3 10 x 1 2x 2 +x 3' >5 X1, x2, x30 MTE-12. 8

9 (b) R*. f4itf117t 31tziaT* 1:1-Ri -TT a-ttfrim ct) mu( 5 t' *err chi t" rcw p ia t 13T ch411r4i 4)14 ciit a1 1-31d-rr t 4)14 ch d1w.1 1ft 3f-d7 t argin.1,4-1=a ct,i4 f*--d--4 ii Rqi IlzfT t I 411 ch 'OK E F G A B C ) chi 410 m cnr eirdi TIR 116 * pi ff-d rf kp-14 TT tr? 4. (a) t Iteffff TECT-4-0 * fm f1 fcirict 7 R'W NOT*' alttrt VTIW t I V NR FW artwit t TIT I WI4 *-69' -N-fqdc1-11 w11--a-rfaci-1131t4kiwt*wri Vf-A7 I MTE-12 9 P.T.O.

10 (b) TrkF (x, I, x2 + y2 y2 x Tf '1(51F-cii 3-9-r- err r 3T-a t? d7r E --1-r--A7 5. (a) li of P1*P21 cbot 5 I Pi T ferra re-r, 711-TMFWT4 1Q-K 2' 3i"1 a ct.),(4 Ri 5 t, P W 7-*-T1 fert4 rc,ni 2 -wt4 rc-ni 3 N.z1 aril 7R f-eqT tr-0 t * 2 4d't t.51 -ch11-1 ctlk qtm 40 ';ft mrci Tag, 31-A-W Hirc,Ri chof 60 'Eft m ro v97 *SFA-W 4d( 50 1=ft if'cr Trmt t I P1 * )t-)i-ri: 3T. *4T. ti md Tuft Trni T)- c -a-r417qt;ert 'ftw turit9- TFRza T1-wTur --1-f-A7 (b) r7---ircirocr -urch--14 rcirtt -1f-A 7 5 3t{ ath rstii sf-r -L5r-d-14 1-r-wrzrT gild rtg ciisl A ro (111'1 B MTE-12 10

11 its10141 A 3-t B LPP fit4 -F:t 5 tracmui wl* : fistoisl B itsmi4 A d7r-4 k m4 44 ch 11 fqfq (1 tsici 1T wr-ferr rtfw ter' vtrrff Wr-4R : S2 S3 D 2 D3 D ' ai bj (c) TIR A = [ ' B = 5 3 ath C= ABC, CBA, BAC 34TBCA 414 ch 1-14 ijui-vt)ri trilitritrff t ( 1179T Renirid c Atmd 3174 WIFTR 1 MTE P.T.O.

12 rrd 11 "4 ch -1=4.21-9'1c ct) affiff k chtt oi --ffttr I f*-t3-1-9u T WF 31-dialg *I LPP-r3Trtratttg-ifaT#Tra-t1T-*iti m 1 n TI1TFIT*W1:1 4t.173TT-4-64tse.tim+n-lt t : (e) lifq7 1 o 'o 1 at,o, - o 0 o - MTE-12 12

BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination June, 2015

No. of Printed Pages : 12 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination June, 2015 MTE-12 ELECTIVE COURSE : MATHEMATICS MTE-12 : LINEAR PROGRAMMING Time : 2 hours Maximum Marks : 50 (Weightage