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IGNOU MTE-12 - Linear Programming

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IGNOU MTE-12 Code Details

  • University IGNOU (Indira Gandhi National Open University)
  • Title Linear Programming
  • Language(s) English
  • Code MTE-12
  • Subject Mathematics
  • Degree(s) B.Sc.
  • Course Core Courses (CC)

IGNOU MTE-12 English Topics Covered

Block 1 - Basic Mathematics and Optimisation

  • Unit 1 - Basic Algebra
  • Unit 2 - Inequalities and Convex Sets
  • Unit 3 - Optimisation in two Variables
  • Unit 4 - Optimisation in more than two Variables

Block 2 - Simplex Method and Duality

  • Unit 1 - Standard Forms and Solutions
  • Unit 2 - Simplex Method
  • Unit 3 - Primal and Dual
  • Unit 4 - Duality Theorems

Block 3 - Special Linear Programming Problems

  • Unit 1 - Transportation Problems
  • Unit 2 - Feasible Solutions of Transportation Problems
  • Unit 3 - Computational Method for Transportation Problems
  • Unit 4 - Assignment Problems

Block 4 - Games Theory

  • Unit 1 - Games with Pure strategy
  • Unit 2 - Games with Mixed Strategy
  • Unit 3 - Graphical Method and Dominance
  • Unit 4 - Games and Linear Programming
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IGNOU MTE-12 (January 2023 - December 2023) Assignment Questions

1. Which of the following statements are true and which are false? Give a short proof or a counter-example in support of your answer. a) In a two-dimensional LPP solution, the objective function can assume the same value at two distinct extreme points. b) Both the primal and dual of an LPP can be infeasible. c) An unrestricted primal variable converts into an equality dual constraint. d) In a two-person zero-sum game, if the optimal solution requires one player to use a pure strategy, the other player must do the same. e) If 10 is added to each entry of a row in the cost matrix of an assignment problem, then the total cost of an optimal assignment for the changed cost matrix will also increase by 10. 2. a) Solve the following linear programming problem using simple method: b) Using the principle of dominance, reduce the size of the following game: Hence solve the game. 3. a) Find all basic feasible solutions for the following set of equations: b) Examine convexity of the following sets: 4. a) Solve the following linear programming problem by graphical method: b) Find the dual of the following LPP: 5. a) Find the initial basic feasible solution of the following transportation problem using matrix-minima method: Also, find the optimal solution. b) Solve the following game graphically: 6. a) A firm manufactures two types of products, A and B, and sells them at a profit of ₹2 on type A and ₹ 3 on type B. Each product is processed on two machines M1 and M2. Type a requires one minute of processing time on M1 and two minutes on M2; type B requires one minute on M1 and one minute on M2. The machine M1 is available for not more than 6 hours 40 minutes while machine M2 is available for 10 hours during any working day. Formulate the problem at LPP. b) Solve the following assignment problem: 7. a) The following table is obtained in the intermediate state while solving an LPP by simplex method: Check whether an optimal solution of the LPP will exist or not. b) Write the LPP model of the following transportation problem: c) Find the range of values of p and q which will render the entry (2, 2), a saddle point for the following game: 8. a) Test the convexity of the following sets: b) Determine all the basic feasible solutions to the equations Identify the degenerate basic feasible solutions. 9. a) Let Compute AB, BC, AC, if they exists, otherwise, give reason for their non-existence. b) Solve the following LPP: c) Find the saddle point (if exists) in the following pay-off matrix: Also, find the value of the game. 10. a) Determine an initial basic feasible solution to the following transportation problem and hence find an optimal solution to the problem: b) Find all values of k for which the vectors are linearly independent.

IGNOU MTE-12 (January 2022 - December 2022) Assignment Questions

1. State if the following statements are true or false. Justify your answer. a) The intersection of a finite number of convex sets is not convex. b) There exists an extreme point of the common set of feasible solution to a L.P.P. which is not a basic feasible solution. c) If value of the 2× 2 matrix game d) If 10 is added to each of the diagonal entries of the cost matrix in an assignment problem, then the total cost of an optimal assignment for the changed cost matrix will increase by 10. e) For the matrix game (A2 , B1) is a saddle point, but (A2 , B2) is not a saddle point. 2. a) Test for convexity the following sets: b) Determine all the basic feasible solutions to the equations Identify the degenerate basic feasible solutions. 3. a) A firm manufactures pills in two sizes A and B. A contains 2 grains of aspirin, 5 grains of bicarbonate and 1 grain of codeine. B contains 1 grain of aspirin, 8 grains of bicarbonate and 6 grains of codeine. It requires at least 12 grains of aspirin, 74 grains of bicarbonate and 24 grains of codeine for providing immediate effect. It is required to determine the least number of pills a patient should take to get immediate relief. Formulate the problem as a standard LPP and solve it graphically. b) Write the dual of the following LPP: Maximize z = 2x 1+ x2 4. a) Let Compute AB, BC, AC wherever defined. If you think some of these are not defined, give reasons for your answers. b) Check whether the following LPP has a feasible solution using two-phase method 5. a) Determine an initial basic feasible solution to the following transportation problem using matrix minima method and hence find an optimal solution to the problem: b) Find all values of k for which the vectors are linearly independent. 6. a) A departmental head has four subordinates, and four tasks to be performed. The subordinates differ in efficiency, and the tasks differ in their intrinsic difficulty. His estimate, of the time each man would take to perform each task is given in the matrix below: How should the task be allocated, on to a man, so as to minimize the total manhours? b) Using dominance solve the game whose pay off matrix is 7. a) Solve the following 2×3 game graphically: b) Consider the following transportation problem: i) Is this transportation problem balanced? Give reasons for your answer. ii) Obtain a basic feasible solution to the above transportation problem by North-West Corner method. 8. a) Solve by simplex method for following linear programming problem: b) Show that the set of vectors from a basis for E 3. 9. a) Using the initial basic feasible solution for the transportation problem given below, find an optimal solution for the problem. b) Test the following set for convexity 10. a) The following table is obtained in the intermediate stage while solving an LPP by the simplex method. Discuss whether an optimal solution will be exist or not. b) i) Formulate the dual of the following problem: Ii) Check whether (2 ,2, 2) is a feasible solution to the primal and is a feasible solution to the dual. iii) Use duality to check whether (2 ,2, 2) is an optimal solution to the primal.
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