Linear Programming and Applications — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

(b) Convert the following Game problem into a pair of Linear Programming Problems for Player A and Player B. (c) Solve the following cost Transportation Problem : NEP-UGCF 2022 - B.Sc. (H) l. (a) Solve tha following Linear programming problem by Graphical Method: MinimizeZ:_xr_3x.-subject to xr + x, S 6 (b) Find all the Basic Feasible Solutions for the following system of equations xl+x.+2x,-xo=6 (c) Define a convex set and use its methematical definition to prove that the following set S is a convex set : 1 (a) Consider the following problem: (c) A company has four machines to be installed at five vacant places available. Each machine can be installed to one and only one place. Find the optimal solution of the Assignment problem and the total minimum cost with the cost matrix given below. Also find which placc remain vacant: 6. (a) Define a Saddle point. The pay-off matrix of a game is given below. Find the solution of the game of Player A and Player B. Is it a Fair Game? 5. (a) Find the optimal solution of the Assignment Problem {,ith the following cost matrix. Also find an alternate optimal solution, if any Given a Basic Feasible Solution xa = B 'b, if for any column a* of A not in B, z* - cu t 0 and y11 S 0, i : 1,2,...... ..., m; then prove that the problem will have an unbounded solution. (b) For the following cost minimization Transportation Problem, find initial basic feasible solution by using North-West comer rule, Least Cost method and Vogel's approximation method. Compare the three solutions (in terms of cost). (b) Solve the following Linear Programming Problem by Simplex Method : (c) Solve the following Linear Prograrnming Problem by Big-M method Minimize Z:-xr-x, subject to x, - x, - x,, : I (a) Solve the following Linear programming problem by Two Phase Method: Minimize Z=3x,- 3xr+x, subject to x, * 2x, - x, ) 5 (b) Prove that to every extreme point of the feasible region, there corresponds a basic feasible solution of the l-inear Programming problem: Ivliuiniize z - ux subject to Ax = b, x 2 0. (c) Solve the following Linear programming problem by Simplex Method: Find an alternate optimal solution, if it exits. How DJ-eDJ/ opt;,tat foasibte solutions docs the problem has? 4. (a) Write the Dual of the following Linear Programming Problem so that dual variables and the requirement vectors are non-negative : (b) Prove that Dual of Dual is Primal. same for the following problem: Maximize Z=-xr*3x, subject to 2x, - 5x, ( 8 Verify the (c) Using Complementary Slackness Theorem, find optimal solutions of the following Linear Programming Problem and its Dual.