BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination June, 2017

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1 No. of Printed Pages : 12 MTE-121 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination June, 2017 ELECTIVE COURSE : MATHEMATICS MTE-12 : LINEAR PROGRAMMING Time : 2 hours Maximum Marks : 0 (Weightage : 70%) Note : Question no. 1 is compulsory. Do any four questions out of questions no. 2 to 7. Use of calculators is not allowed. 1. State which of the following statements are true and which are false. Give reasons for your answer with a short proof or a counter-example. x2=10 (a) (b) (c) The LPP, maximizing z = - x 2 subject to x1 + x2 1, 0.xi + x2 0 and xi 0, x2 0, has no feasible solution. If a negative value appears in the solution column (xb) of the simplex method, then the basic solution is unbounded. If the variables in a primal problem are all greater than, or equal to zero, then the variables in the dual problem would be less than or equal to zero. MTE-12 1 P.T.O.

2 (d) The value of the game I II III IV I II 0 6 III is 6. (e) The minimum number of lines covering all zeros in a reduced cost matrix of order n in an assignment problem can be at least n. 2. (a) Formulate the dual of the following LPP : Maximize z = 2x 1 + x 2 subject to the constraints x 1 + x 2 10 x + 3x 2 6 2x 1 8 x 2 0 and x 1 unrestricted. The dual must have exactly two constraints and three variables. (b) Solve graphically the following LPP : Maximize z = 3x 1 + 2x2 subject to the constraints - 2x1 + x2 1 x1 _ 2 xi + x2 3 x1, x2 0. MTE-12 2

3 3. (a) Use simplex method to solve the following LPP : Maximize z = x 1 + 3x 3 subject to the constraints x + 3x < xi + x2 x1, x2, x 3 0. Also, compute one more alternate optimal solution if it exists. (b) Solve the following game graphically : (a) Five men are available to do five different jobs. From past records, the time (in hours) that each man takes to do the job is known and given in the following table : Men A B C D E Jobs II III Find the assignment of men to jobs that will minimize the total time taken. Also, compute the time. MTE-12 3 P.T.O.

4 (b) Solve the following LPP using the two phase simplex method : Maximize z = 3x 1 + 2x2 + x3 subject to the constraints - 3x 1 + x 3 = 8-3x 1 + 4x 2 + x 3 = 7 x1, x2, x3?_ 0.. (a) Use dominance principle to solve the game whose pay-off matrix is given by Player A I [ II III IV I 6 2 Player B II III IV Also, find the value of the game. 6 (b) A Chemistry laboratory uses raw material I and II to produce two domestic cleaning solutions A and B. The daily availability of raw materials I and II are 10 and 14 units respectively. One unit of solution A consumes 0. units of raw material I and 0.6 units of raw material II and one unit of solution B uses 0. units of raw material I and 0.4 units of raw material II. The profit per unit of solutions A and B are 8 and 10 respectively. The daily demand for solution A lies between 30 and 10 units and that for B between 40 and 200 units. Formulate the above data as LPP. 4 MTE

5 6. (a) For the following problem in the game theory, formulate the problem of finding the optimal strategies for player A and player B as linear programming problem : Player B Player A (b) Find the initial basic feasible solution of the following transportation problem to minimize the cost using matrix-minima method : Warehouses W1 W2 W3 W4 Capacity F Factories F F Demand Hence, find the optimal solution. 7. (a) Consider : S = {(x, E R2 I xy x 1,y 1, x 0, y 0}. Is S a convex set? Give reasons. Determine the convex hull of S and describe it using mathematical inequalities. 4 MTE-12 P.T.O.

6 (b) Consider the following transportation problem : S 2 S 3 b.-> D 1 D 2 D 4 D a. 4, If u 1 = - 1, u2 = 20, u3 = 0, v 1 = 30, v2 = 60, v3 = 4, v4 = 2, v = 0, then find the corresponding basic feasible solution of the problem. Is this solution optimal? Give reasons. 6 MTE-1 2 6

7 71.t ) S1144 T4, 2017 ' /WtW t)1.4 : 1 11j ICI "crgi.t.t.-12 : (Slch 1, HAN : 2 Wu2 31-r 3 : 0 (mf r : 70%) #. 1'<i13 /AT PI 4 W7? / cic-,dej 3Te7T 4,4 3.7-q-0 07* / 1. odiv 4) PIH orgd war-4 Hrect 37-R-R7 ITi-rwi 3trIA Nc- qwuf 3TT 371k chlkuf.-d-qr I x2=10 () -1briT4 NCI I, z = - x2 f 3TR1-*--di#W{71 *f.n7 77k xi + x2 S 1, 0.x 1 + x2 0 at xi > 0, X2 0 Tide t (13- ) -cr-wil-r MT* -1:Trni (3( B) e4r 7rtutic4l.b t, 3Trtrrft 3Tqft-oz t (Tr) qr 3Tr-cr TrTr-fzrr *Rift A. 331W t, Trgr-FErr * * -qfr * tsklel.k Zif 7LP:I err 1,--e4 I MTE-12 7 P.T.O.

8 (Er) LAM I II III IV I II 0 6 III IV ftzid-4 TrA-fErr A, n lirizi1 A k-ei IRA n Tr*-t t I ff-d14 2. () 111 1").71W4 TII:RzFT *"1 Tkrqff z = 2x 1 + x 2 3irWallt*TIT*r-41 1:1 4-1R161 -&T-4611* 3T- Th x 1 + x 2 <- 10 x 1 + 3x 2 > 6 2x 1 < 8 a x 2 >04R x 1 347r-d-A-u -4 - gl.t) th-r- 3Tr-4q-zr- (1g) PHRIRgd tgra-4 Tri:R=41 7rfrer 'MT z = 3x 1 3-TRT+7141-*-TuT 1:1 4-11CIRVI - 2x 1 + x 2 <- 1 x 1 < 2 x + x < x 1, x2 0. MTE-12 8

9 3. trap:17 ITIWTE T 1-4RT z x 1 + 3x3 $it 3TRWdlit*TIT*fr7R PiHRiRgi x 1 + 3x 3 < 10 x + x < 1 2 X1' x2, x3 0. -> c..on PdC1 1-1 ca t, cb orgct 9TOzr fq-nt (.1*'1.NR : l () ITN fat-ff cnk4 ate-01 zrr-o-ati t Ivi a1 3{pft Asstcb(4 11) tin t 311T PHRIRgi fffr MW1 14 igen TRIT : 31-r-4-Lft A B C D E 161 IV KFTRII 1-44T4 fdf fff-zrr Trzrr Trrrzr * I TP:14 1-1RchRi Ci WA-R MTE-12 9 P.T.O. V

10 i 4-'1RiRgd -srliztf TP:F;Eff t- -crul ITT -rea : z = 3x 1 + x3 T t41* 3x 1 + x 3 = 8-3x 1 + 4x 2 + x 3 = 7 x1, x2, x 3 > 0.. -rrjum wrirt cht, PiHRi (1 1IlH Atr-A7 : Rgc-114 B I II III IV I fil(11 9 A II III ITR ift 411c1 Arr-4R I 6.1-) kwlelpich 1e11 1 R11(11 Trchr R A 311T B *114 rml ctri34 c 13I II w 7-41-rr. 7-d1t I chi A II4t 7r-oTig-d-r 9b 4-RI: cf-oeil I 14ffzR A t, T*-11 ch-cd 411( W4 A chi Trra II 0.6c*--f TE-4't -4 A fa o > T*-r i ch -cd HIM I 0. 3 ch-c MO II 0.4 T*-4.frth t I fdazr A 3 t B - r UPI -ff13-1 stari: 8 3i 10t I id-07r A *1 *TT 30 A 10r+7z ( Wez:14 B trt- Trirr 40 34i 200 T*-rzil *t ti 34-*71 tgrq-4 TrIR:ziT 7 I 4 MTE-12 10

11 6. () "RTZT- 4t1 Hrrl 1 1 wrzrr 1(goi A 3 Rgc-10 B -1Z-R tra lit E Trlwrrq I:FR:Err kr-1-u 4tF-4l : 6(10 B I A :14--ci rift-4f chkui chk:.) 1 i-mac 14N -9-Frmi- 3Trti-rft Th-rff : Trl-q-rrr W1 W2 W3 W4 klirdt F ohlkn I F F TaTT 'Fr NchR, 41Ic1 QN7 I 7. PIHRifigd : S=Rx,y)E112-47IT S t&ch 31---a-rju Trg-*--zr? ctaul tf--ar ki 4-11clich 4-Ild W77 TriVrt4 34Trrgw[311 -FOr i-r-a7 I 4 MTE P.T.O.

12 (t) e_lcrtrit TrI:RErr S i S 2 S 3 D D 2 D 3 D 4 D a b. > J e4r u, = 1, u2 = 20, u3 = 0, v 1 = 30, v2 = 60, v3 = 4, v4 = 2, v = 0 *, crit.7* 34-1 cr 31mt crii Tff1=1.? chrui tnr I 6 MTE ,000

BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination June, 2013 ELECTIVE COURSE : MATHEMATICS MTE-12 : LINEAR PROGRAMMING

No. of Printed Pages : 12 MTE-12 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination June, 2013 ELECTIVE COURSE : MATHEMATICS MTE-12 : LINEAR PROGRAMMING Time : 2 hours Maximum Marks : 50 Weightage