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.
(d) The value of the game I II III IV I - 3 1 II 0 6 III -4-2 4 7 2 6 8 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. (a) Use simplex method to solve the following LPP : Maximize z = x 1 + 3x 3 subject to the constraints x + 3x < 10 1 3 - xi + x2 x1, x2, x 3 0. Also, compute one more alternate optimal solution if it exists. (b) Solve the following game graphically : -2 0 3-1 -3 2-4 4. (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 2 9 2 7 1 6 8 7 6 1 4 6 3 1 3 9 1 4 2 7 3 1 Find the assignment of men to jobs that will minimize the total time taken. Also, compute the time. MTE-12 3 P.T.O.
(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 6 4 6 4 4 6 4 2 IV 2 6 2 10 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-12 4 4
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 1 2 2 Player A 0 2 6 4 2 6 (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 1 7 3 8 300 Factories F2 4 6 9 00 F3 3 6 4 200 Demand 200 300 400 100 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.
(b) Consider the following transportation problem : S 2 S 3 b.-> D 1 D 2 D 4 D 40 0 30 10 70 100 80 7 6 20 30 60 4 0 40 1 2 20 10 20 a. 4, 20 20 0 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
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