No. of Printed Pages : 11 MTE-12 BACHELOR'S DEGREE PROGRAMME Term-End Examination C 3 ti June, 2012 0 ELECTIVE COURSE : MATHEMATICS MTE-12 : LINEAR PROGRAMMING Time : 2 hours Maximum Marks : 50 Note : Question no. 1 is compulsory. Do any four questions out of 2 to 7. Calculators are not allowed. 1. Which of the following statements are true? 10 Give reasons for your answers : (a) Every convex set has at the most finitely many extreme points. (b) If A and B are 3 x 3 matrices with IAN = 0, then at least one of the matrices A or B has rank less than three. (c) (d) (e) If the primal LPP is infeasible, the dual is also infeasible. In an optimal solution of a balanced transportation problem with 3 sources and 4 destinations, it is possible to have 7 routes over which a positive quantity is transported. The pay off matrix of any game can have at most one saddle point. MTE-12 1 P.T.O.
2. (a) Use the graphical method to solve the 6 following LPP : Max 2x1 + 3x2 subject to x1 + xi x2?-0, x2.?; 0x 1.20 and Ox2.12. (b) Find the dual of the following LPP : Max subject 2x1 + x2 subject to xi +5x25_10, +3x2.- 6, 2x1 + 3x2 = 8, x2?-0 and xi unrestricted. 4 3. (a) Solve the following LPP by simplex method : 5 Max 2x1 +3x2 + x3 subject to 7x1 + 3x2 + 2x3 5 4x1 + 2x2 x3 2 xi, x2, x3 > 0 (b) Write the mathematical model of the 5 following transportation problem : D1 D2 D3 D4 Supply 51 3 7 6 4 5 S2 2 4 3 2 3 S3 4 3 8 5 3 Demand 3 3 2 2 Also, find an initial basic feasible solution using North-West corner method. MTE-12 2
4. (a) Find an initial basic feasible solution using 6 matrix minima method and hence solve the following transportation problem using u v method. From To A B C Available I 6 8 4 14 II 4 9 8 12 III 1 2 6 5 Demand 6 10 15 (b) Find all the basic solutions of the following 4 equations : xi + X 2 + 2x3 = 5 2x1 X 2 + X3 = 4 Which of the solutions are feasible? Justify your answer. 5. (a) Solve the following game graphically : 5 Player B [0 5-3 3 Player A 3 2 (b) Solve the following cost minimising 5 assignment problem : I H Ill IV A 20 28 19 13 B 14 29 15 27 C 38 19 18 15 D 20 27 25 11 MTE-12 3 P.T.O.
6. (a) Prove algebraically that the set 5 S = {(xi, x2) /2xf + x2 5._ 4} is a convex set. (b) Use the method of dominance to reduce the 5 size of the following game, and hence find its solution : 4 1 2 4 2 3 4 3 2 1 5 2 3 2 5 3 7. (a) Players A and B simultaneously call out 5 either of the numbers 19 or 20. If their sum is even, B pays A that number of rupees. If the sum is odd, A pays B that number of rupees. What is the pay off matrix for this game? Find the optimal strategies for both the players. (b) A company has two grades of inspectors, 5 I and II, who are to be assigned for a quality control inspection. It is required that at least 2000 pieces be inspected per 8 - hour day. A grade I inspector can check at the rate of 40 per hour with an accuracy of 97%. A grade II inspector checks at the rate of 30 pieces per hour with an accuracy of 95%. The wage rate of a grade I inspector is Rs. 5 MTE-12 4
per hour while that of a grade II inspector is Rs. 4 per hour. An error made by an inspector costs Rs. 3 to the company. There are only nine grade I inspectors and eleven grade II inspectors available in the company. Formulate the problem of minimising the daily inspection cost as an I,PP. MTE-12 5 P.T.O.
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(e) fc 31-r--&ff Trd-r-i 3fti-T 31-fqw.icht'llt I 2. (a) F'1+-irciroci LPP arn A 60 : 6 2x1 + 3x2 3-ifq-*71:1t-*77 I 7'qfT x1 +x2530, xl x2 > 0, x2.>- 3, 05x1520 bit 05x2512. (b).tati LPP 711c1 : 4 2x1 +x2 3-1fq-*7711M-711T v T : 7-4f- xi +5x2510, xi + 3x2?- 6, 2x1 + 3x2 = 8, x2 >0 34 x1 13TIXF-d-4RTU I 3. (a) 14-110{47 LPP ch) ittz.11 PART 4 Sri : 5 2x1 + 3x2 + x3 WI 3-Tfq--*---d-rn--*---m 7-4F-T 7x1 + 3x2 + 2X3 5 5 4x1 + 2x2 X3 5 2 xi, x2, x3 > 0 (b) 14-1 Oci o.-1 Trr-R:zu f,trukilq 5 tfur : D1 D2 D3 D4 3Trf4 S I 3 7 6 4 5 S2 2 4 3 2 3 S3 4 3 8 5 3 1Ttf 3 3 2 2 alit- 14 -NN bra qiicf "4-177 I MTE-12 8
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